.ARli;boos/library/mimeo.archive/ISMS_1966_472.pdfMULTI-STAGE SAMPLING ON SUCCESSIVE OCCASIONS WHERE FIRST-STAGE UNITS.ARli; DRAWN WITH UNEQUAL PROBABILITIES AND HITH REPLACEMEET by

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MULTI-STAGE SAMPLING ON SUCCESSIVE OCCASIONS

WHERE FIRST-STAGE UNITS .ARli; DRAWN

WITH UNEQUAL PROBABILITIES AND HITH REPLACEMEET

by

Niyom Purakam and. John C. Koop

Institute of StatisticsM:l.meograph Series No. 472April, 1966

•

,

ABSTRACT

PURAKAM, NIYOM. Multi-Stage Sampling on SUccessive Occasions where

First-Stage Units are Drawn with Unequal Probabilities and with

Replacement. (Under the direction of JOHN CLD:lENT KOOP).

A multi-stage sa~ling design, particularly intended for large

scale sample surveys on successive (or repeated) occasions is developed.

The sampling design is general in the sense that the probabilities of

selecting units (for the preliminary first-stage sa~le) are arbitrary.

Each of these first-stage units is drawn with replacement. The

technique of partial replacement of first-stage sa~ling units is based

on the order of occurrence of these units. The partial replacement

technique is developed to meet two basic objectives:

(i) To spread the burden of reporting among respondents which may

be expected to help in maintaining a high rate of response.

(ii) To enable the sampler to take advantage of the saItij?ling design

in the reduction of sampling variance of several estimators proposed.

Several ways of utilizing the past as well as the present informa

tion from the sampling design to estimate the total, and the change in

total of a population characteristic of interest, are presented. The

nature of the gain in efficiency from using the four different forms of

estimators in estimati~g the total, and the change in total, is explored.

The comparisons of efficiency among the estimators wherever possible,

are given under certain assumptions simiiar to the assumption of Second

Order or Weak Sense Stationarity· usedin conventional time series

analysis.

•

,

t,

The estimation theory is covered in detail for two-stage sampling

on two successive occasions. The extension to higher stage sampling on

more than two successive occasions is sufficiently indicated, In all,

the reduction in the variance of an estimator whenever achieved, is in

the total variance namely, the between first-stage units variance plus

the within first-stage units variance, and so on if there are more than

two stages of sampling.

..

iii

ACKNOWLEDGEMENTS

To Dr. J. C. Koop, the chairman of my advisory connnittee, I wish

to express my sincere appreciation and thanks for his guidance during

my graduate work here. It was mainly through his suggestion and

direction that the work reported in this thesis was completed.

My thanks are extended to other members of my advisory connnittee:

Dr. R. G. D. Steel, Dr. J. Levine, Dr. N. L. Johnson and Dr. B. B.

Bhattacharyya for their useful suggestions and criticisms of th~ draft.

I am also very grateful to The Agency for International Development

for the financial support which enabled me to pursue graduate study

in this country.

For their respective shares in the work of typing the draft and

the final version of this thesis, I wish to express my heartfelt thanks

to Mrs. Ann Bellflower and Mrs. Selma McEntire.

Finally, to my father who had to struggle all his life to support

my early education, but who never saw the result at this stage, this

thesis is humbly dedicated •

.. iv

TABLE OF CONTENTS

LIST OF TABLES • . . . • 0 • 0 0 • • • • • vii

Page

19 ~ • •. . .. .INTRODUCTION •1.

o • • • •1.1. Basis for the Present Investigation1.2. Nature of the Problem • • • • • • • 0 • •

• • 12

13

6

· .

. .. . .000000 •• 02 • REVIEW OF LITERATURE • • • •

3. A PROPOSED SAMPLING DESIGN

3.1. Description of a Method of Partial Replacement ofFirst-Stage Units ••••••• • • • • • • • • •• 12

3 .2 • Advantages of the Proposed Scheme of PartialReplacement of First-Stage Units • • • • • • • • • • 14

3.3. Specification of Probability System and the Methodof Selecting Sub-Units • • • • 16

4. ESTD1ATION THEORY • 0 • 0 • • 21

4.3.1.4.

4.3.1.5.

4.3.1.1.

4.3.1.2.

4.3.1.3.

4.1.4.2.4.3.

Introductory Remarks • • • • • • • • 21Estimation of Total for the First Occasion • • • 23Estimation of Total for the Second Occasion • • 27

4.3.1. A Linear Composite Estimator • ••• 28....

Expected Valu; of 2T •••• ,. " 29

Variance of 2~ • • • • • • • • • 30

Note on the Estimation of""-

Var (2~) • • • • • • • •• ,.. 42~

Efficiency of 2T • • • • • • • • 43

Choice of Q in the LinearConu;>osite Estimator ••• • • 47

4., .1.6. Choice of Ii ...•...•.• 524.,.1.7. Simultaneous Optimum Values of

Q and IJ. in the LinearComposite Estimator • • • • • 54

40,.1.8. Expression for the MinimumAttainable Variance of the ""Linear Conu;>oSite Estimator 2t!'. 56

4.3.2. A Modified Linear Composite Estimator '. '. '. 594.3.3. A General Linear Estimator • • • • • • • 66

4 .3 ., .1. Determination of Constants • 674.,.,.2. A Comment on the Form of th~

Estimator 2T* • • • • • • • • 73

v

TABLE OF CONTENTS (continued)

Page

99

76

79

7982

91

93

94949496. . .

4.3.3.3.

4·3.3·5.

Efficiency of the Estimator 2T*. 73Optimum Value of IJ. when the,

Estimator /i* Is Used. ,. .' ,.

4.3.4. A Ratio-Type Composite Estimator ••• • " "~*4.3.4.1. Expected Value of 2T •••. " "

4 4 A*·3. .2. Variance of 2T .•••• ,. " "

4 . l\i\*4.3 •.•3. EffJ.ciency of 2T .,. ••

Estimation of Change in Total between the First and'Second Occasion • • • • • • • • • • . • • • • • • •4.4.1. An Estimator Based on the Linear Composite

Estimator •• • • • • • • • • • • • •4.4.1.1. Estimator •• • ••4.4.1.2. Variance ••••••••.' .' .' .'

~

4.4.1.3. Efficiency of D1 •

4.4.1.4. A Comment about the Gain inEfficiency • • • • • • • •

4.4.1.5. A Remark about the Optimum .Value of Q • • • • • • • • • • 102

4.4.2. An Estimator Based on the Modified LinearComposite Estimator •• • • • • • •• 103

~t4.4.2.1. Variance of D1 •••••••• ,. 103

At4.4.2.2. Efficiencyof P1' P ,••••• " ,.104

4.4.3. A General Linear Estimator .•••••••• 1084.4.3.1. Determination' of Constant· a','

b " C I, d I • • • • • • • • • • 108

4.4.

..

4.4.3.2 • A Comment about the Form of. the,

""*Estimator D1 • • • • .,. ," 111A*4.4.3.3. Variance of D1 ••••••••• 111

4.4.3.4. Efficiency of the Estimator Dr 112

4.4.4. An Estimator Based on the Ratio-TypeComposite Estimator • I' • • • • •• •• ,. 114

A*4.4 .4 .1. Bias of D1

• • • • •• " " 114s*4.4.4.2. Variance of JJ1 • • • ••••• " 115

4.4.4.3. A Remark about Var (~~) •••• 117

vi

TABLE OF CONTENTS (continued)

Page

. . .5. MORE THAN TWO STAGES OF SAMPLING AND MORE THAN TWOSUCCESSIVE OCCASIONS • • • • •• • • . 119

· • 138• 146

• . 119• . 119• • 127• • 128• • 128

General Remarks • • • • • • • • • • • • • • • •More Than Two Stages of Sampling • • • • • • •More Than Two Occasions • • • • • • • • • •

5.3.1. Estimation of Total •••••••••••5.3.1.1. A Linear Composite Estimator5.3.1.2. A Modified Linear Composite

Estimator •••••••••5.3.1,.3. A General Linear Estimator5.3.1.4. A Ratio-Type Composite

Estimator • • • • • • • • • • • 152Estimation of Change in Total • • • • • • • • 156

5.3.2.1. The Estimator Based on theLinear Composite Estimator 156

5.3.2.2. The Estimator Based on theModified Linear CompositeEstimator • • • • • • • • • • • 156

5.3.2.3. A General Linear Estimator • • • 1575.3.2.4. The Estimator Based on the

Ratio-Type Estimator • • • • • 159

Note on the Estimation of Sampling VariancesNote on the Efficiency of' the Estimators when

1Vy :f 2Vy • • • • •••••• • • • • • • •

160

• 163

· .• ••• 160• ••• 161

. . .

. . . . .• • 0 • • • • 165

· . 165

••.•• 167

. .. . .

• • 0 • • • •

o 0 0 0 • • 0

. .• • 0 • 0 0'

. .• • • I)

. .• • • 0. .

8.1.8.2.

6. SUMMARY AND CONCLUSIONS •

6.1. Summary. • •6.2. Conclusions.

7. LIST OF REFERENCES • • • • • • • •

8. APPENDICES • • • • • • • • • • • •

J.

t.

vii

LIST OF TABLES

Page

;.1. Scheme of sample selection, partial replacement andconstituent samples ••.•••••• • • • • ~ • • • 15

4.1

4.2.

111 A

Percent gain in efficiency of 2T over 2T

1Optimum value of Q for ~ ~ 4 and percent

of 2~ over 2fE • • • • • • • • •

. . .gain in effici!3n~y

48

50

4.4.

4.6.

1Optimum value of Q for ~ ~ ~ and percent gain in efficiency,A A ./

of 2T over 2T • • • • • • • • • • • • • • • • • • • • • • 51

1Optimum value of Q for ~ ~ 2 and percent gain in eff1~iency,A A

of 2T over 2T • • • • • • • • • • • • • • • • • • • • • • 52

Optimum value of ~ for Q = ~ and the percent gain in~ A

efficiency of 2T over 2T • • • • • • • • • • • • • 54

* APercent gain in efficien~ of 2T over 2T for simultaneous

optimum values of Q and ~ • • • • • • • • • • • • • • • • 58

. .

•

r

4.7·

4.8.

4.10.

4.11

4.12 .

4.13.

4.14.

4.15·

~I AllPercent gain in efficiency of 2T over 2T for ~ = 2 ' Q = 2.

~

Optimum value of Q to be used in the estimator 2T for1 A

~ = 4 and the percent gain in efficiency over 2T ••••AI

Optimum value of Q to be used in the estimator 2T when1 A

~ = 3' and the percent gain in efficiency over 2T ••~I

Optimum value of Q to be used in the estimator 2T when1 A

~ = 2' and the percent gain in efficiency over 2T

A* A 1Percent gain in efficiency of 2T over 2T for ~ =4 . .Percent gain in efficiency of /r* over /i for ~ = ~

Percent gain in efficiency of 2fE* over /r for ~ = ~ ••••

Optimum value of ~ when the estimator 2T*iS used andA

the percent gain in efficiency over 2T • • • • • • • • •

* AllPercent gain in efficiency of Dl over Dl when ~ = 4 ' Q = 2'

63

64

65

66

74

75

76

78

97

viii

LIST OF TABLES (continued)

Page

4.16.

4.17.

4.18.

~ ~ 1 1Percent gain in efficiency of D1 over D1 for ~ =3' Q =2 98

~ ~ 1 1Percent gain in efficiency of D1 over D1 for ~ = 2' Q =2 99

Optimum value of Q to be used in the estimator ~1. and,~ 1

percent gain in efficiency over D1 for ~ = 4 . . .. . 100

• 101

• • • • 106• • • • It •• • • • • It • • •

~

Optimum value of Q to be used in the estimator D1 and~ 1

percent gain in efficiency over D1 for ~ =3' . • . .Optimum value of Q to be used in the estimator ~1 and

~ 1percent gain in efficiency over D1 for ~ = 2 . . . . . . . 102

At ~ 1Percent gain in efficiency of D1 over D1 for ~ = 2 '

1Q=2

4.20.

4.21.

4.19.

4.22.~,

Optimum values of Q to be used in the estimator D1 and~

percent gain in efficiency over D1 • 0 • • • 107

~Percent gain in efficiency of ;T

1 1~ =2' Q =2 . . . . . . . .

over

. . .";T for. . • 0 • 1;6

• • 146

151

• '1;8

• 14;

• 168

• •

. . .

over

. . . .. .

• .. • 0

The percent gain inefficiency of the estimator~ 1 1 2 (

2T, when Q = 2" ' ~ = 2" and 1VY = '4 2Vy)

~ "Percent gain in efficiency of ;T over ;T for some1 1

assumed values of P1 and P2, for ~ = 3' ' Q = 2 . . . 0 •

~ ~

Tt over T fo; ;. rPercent gain in efficiency of1 1

~=2,Q=2······· •••~ "Percent gain in efficiency of ; T' over ;T for

~ = ~ , Q = t for some values of PI and P2 •••

~* " 1Percent gain in efficiency of ;T over;T for ~ = 2~

2T

5.5·

8.1.

8.2.~ ~

The percent gain in efficiency of 2T over 2T when

Q =~ , ~ =~ and 1Vy =t (2Vy) ••• • ••• • • • •• 169

LIST OF TABLES (continued)

ix

8.4.

~

Percent gain in efficiency of 2T' over

Q=l,ll=l and V =2.(V)2 2 1y Ij: 2y

Percent gain in efficiency of 2~' over

Q=l,Il=1. and V =24

(V)2 2 1y 2y

. . .'"2T when

Page

••••• 170

• • • •• 171

,-L· INTRODUCTION

I:~l. Basis for the Pres'ent Investigation

In continuing sample surveys conducted at regular intervals (e.g.

quarterly) for investigating the time-dependent chaXacteristic of

certain dYnamic populations, it is frequently advantageous to use the

- 1/so-called rotation sampling- technique, Whereby a scheme of partial

replacement of sampling units i~ developed in such a way that the

sampling units to be used will be in the sample consecutively for some

fimte number of occasions, then they Will be replaced by newly selected

units • The replacement is done only to a portion of the sample while

the other portion is retained for the next occasion. To such plans of

sampling were attached different names by various authors, such as

I'sampling on succession occasions with partial replacement of units II

[12], "rotation sampling" [4], "sampling for a time-series" [7],

"successive Sampling" [17].

The main advantages of a technique of sampling where· partial

replacement of units is part of the overall sampling design over one

where there is no partial replacement of units (1:..!., taking a new set

of units, or using the same set of units every time) are as follows.

1. Partial replacement of units in the sample spreads the burden

of reporting among more respondents and hence results in better co-

operation from respondents. This is very iniportant from the standpoint

1/ II II- The name rotation sampling refers to the process of eliminat-ing some of the old elements from the sample and adding new elementsto the sample each time a new sample is drawn. (See Eckler [4]).

:2

of maintaining the rate of response when a human population is studied.

Experiences from many census or survey studies (by complete enumera-

tion or ·sSmpling methods) seem to indicate that the respondents tend

to become uncooperative during the third or fourth visit if the survey

is carried out repeatedly and the same sampling units are used. Even

with full cooperation, the respondents may be unwilling to give the

same type of information time a:f'ter time, or they may be inf'luenced by

the inf'ormation which they give and receive at earlier interviews, and

this may make them progressively less representative as time proceeds.

On the other hand, taking a new set of units every time is obviously

more expensive. The compromise is partial replacement of units.

:2. Partial replacement of' units in the sample permi-ts the use of

data from past samples to improve the current estimate of population

characteristics of interest. This can be accomplished by some appro-

priate methods of estimation which takes advantage of past as well as

present information to provide an estimate for the present occasion.

This theoretical advantage is perhaps the most important reason for

using partial replacement of units technique when we have to deal with

time-series characteristics.

A large scale sample survey such as a national sample survey, is

usually carried out in more- than one-stage to reduce time, labor and

costs. The new approach here is that the technique of partial replace-

ment of units can be applied to the units at any stage in the sam,pling

process. However, the U. S. Bureau of Census, in applying the technique

to various periodic sam,ple surveys such as the Current Population

•

3

Survey, the Monthly Retail Trade Survey, the Monthly Accounts Receivable

Survey, etc., partially replaces the last-stage units.

It is of theoretical and of practical interest to consider a some

what different scheme of partial replacement of units, where, in the

multi-stage sampling process, the partial replacement of units is

carried out at the first-stage. At the second-stage (or succeeding

stages, if more than two stages of sampling is used) the sampling may

be carried out in any appropriate fashion as the practical situation

may demand. The above problem seems to have received no attention so

far except by Tik.kiwal [11] in a different context.

1.2. Nature of the Problem

For the development of the theory underlying the proposed tech

nique of partial replacement of units in multi-stage sampling, we

w.i.ll fornmlate the problem as follows.

Consider a population whose characteristics of interest change

With timeJ for exampleJ the number of persons employed or unemployed

in the labor force of a country. These numbers are known to vary from

season to season. It is desired to conduct a large scale sample survey

to estimate such time-dependent characteristics of this population

periodically (say quarterly for a total period of 2 years) using a

multi-stage sampling plan. Also, in order to minimize some undesir

able effects resulting from interviewing the responsent repeatedly

over a long period of time, we seek some replacement techniques in

such a way that our sampling operation will utilize units to be in

Cluded for interviewing only for appropriate number of occasions.

•

4

Then replace them after some occasions by units not selected for inter-

viewing before. More specifically, suppose we are going to use a two

stage Samplini/ design, where, in our population, there are N definable

first-stage units:

t~, ~, , uJEach first-stage unit ui ' (i=1,2, ••• ,N) contains Ni second

stage units. For example, if the first-stage units are taken to be

villages or towns, the second-stage units might be households in those

villages or towns. In case of two-stage sampling, these second-stage

units if they are selected, 'Will actually be visited an.d interviewed

by the enumerators.

In the usual non-rotation sampling method, a sample of n units

out of the N first-stage units 'Would be selected from the population;

after that the procedure is to select ni second-stage units out of the

Ni·found in each selected ni • Hence, the total sample size under thisn

sampling scheme is E ni • If the sampling is done quarterly for 2i=l

years and those units are used every time, we will have to repeatn

visiting those E ni units 8 times. On the other hand, to selectn i=l

a new set of E ni units every quarter may be too costly. If thei=l

partial replacement techI)ique such as the one used by the U. S. Bureau

of the Census, is incorporated in the above two-stage sampling plan,

g/This case is chosen for illustration purposes. Later, we willconsider more tha.n two 3tages of sampling. Also, we 'Will not consider stratification because it does not make any difference in thedevelopment of our scheme of partial replacement of units in thesample •

a-

5

the selected second-stage units will be partially replaced on every

occasion. However, the same first-stage units are still being used on

every occasion. In sequel it will appear that such a sampling design,

using any appropriate estimator, brings about a reduction in variance

only in the within first-stage units part of the variance, while the

major contribution to the total variance of the estimator (!.~., the

between first-stage units part of variance) is not reduced.

Considering the fact that it is the between first~'stage units

part of the variance that contributes most heavily to the total vari

ance in many sub~sampling situations, a sampling design and estimation

procedure which reduces this variance would be desirable. The problem

then to be taken up in this thesis is "How would a sampling design

which reduces the total variance of an estimator be obtained? Further,

what is the appropriate estimation theory?" It is the purpose of this

thesis to investigate the above problem.

2. REVIEW OF LITERATURE

The first attempt to utilize the information obtained from

previous. samples to· improve the estimate for the current occasion

seems to have been made by Jessen [8]. To estimate the population

mean of a characteristic of interest, he conducted a survey on two

successive occasions. On the first occasion, a simple random sample

was drawn (in one stage). On the second occasion, he replaced a part

of the sample drawn on the first occasion, while the remaining portion

was retained for matching. 0 n th es econ d occasion, he obtained

two estimators which were correlated. One was the sample mean based

on the new units only, and the other was a regression estimate based

on the units which were kept for matching, and the overall sample mean

obtained from the units observed on the first occasion. He thus

obtained a new estimator by a linear combination of the two estimates

which he claimed to be a minimum variance linear unbiased estimator

of the population mean on the second occasion.

The form of the estimator used by Jessen is

I

Y. =Q y,' + (l-Q) y~2 2u c:;m

where

Y~u =Y2U =the mean of unmatched portion on the

second occasion, and

6

(2.2 )

In (2.2)

Y2m = the mean of the matched portion on the second

occasion.

Ylm =the mean of the matched portion on the fi~st

occasion.

Yl = the mean of the whole sanr,ple on the first

occasion.

" II dThe regression coefficient b is assume to be known.

1

7

Q =1 + 1

1

1 + 1

The extension of the theory to more than two occasions, also

confined to unistage simple random sampling, was made by Yates [20].

Yates presented the estimation theory which may be viewed as a

generalization of Jessen's theory. In addition, in the development

of the estimation theory for more than two occasions, Yates assumed

that the correlation between the same sampling units for the observed

characteristic on two different occasions is of an exponentially

decreasing type, Le., the correlation between observations made on

.,.

82

one occasion apart is p, two occasions apart is p , three occasions

a.part .is p3 and so on. Yates further assumed that the variances and

covariances did not change with time, i. e.

Cov (Yj'Yh ) = Cov (Yj+k' Yh+k )

where j, h, j+k and h+k denote the jth, hth, (j+k)th and (h+k)th

occasion respectively.

Patterson [12] further extended the theory given by Yates and

derives a necessary and sUfficient condition for a linear unbiased

estimator to be a minimum variance estimator 0 Patterson considered

also types of correlation patterns other than that given by Yates.

Similar theory for unistage simple random sampling is also discussed

in [1] and [15] with some slightly different approaches. other

contributors to the theory of sampling on-successive occasions were

Tikkiwal [16], [17] and Eckler [4] who developed specific schemes of

partial replacement of units and presented·the theory relevant to

their proposed sampling plans. Eckler was instrumental in introducing

the term "rotation sam;plingll, actually suggested to him by S. S. Wilks 0

In 1954, Hansen ~ a1. [6] redesigned the Current Population

Survey (c oP 0 So), from which information on employment, unemployment

and other related socio-economic data are compiled monthly. One

feature of sub~samplingwhich has an important bearing on the estima-

tion theory introduced in the new sample involves a scheme of partial

replacement or rotation of sampling units at the last sta.ge~ This

• 9

sampling technique was primarily intended to avoid a decline in

respondent cooperation (which may happen when the same unit is repeat-

ed1y interviewed ) and to reduce the variances of estimates.

For any given month, the C.P.S. sample is composed of eight sub-

samples or rotation groups. All of the units composing a·particular

rotation group enter and drop out of the sample at the same time. A

given rotation group stays in the sample for four consecutive months,

leaves the sample during the eight succeeding months, and then returns

for another four consecutive months. It is then dropped from the

sample completely.

It was in this kind of continuous sample survey that the so-called

"composite estimation procedure ll was first introduced. The composite

estimator used in this survey is of the form:

I' _ I"I( 11 +' ') + (1 1"1).'Ya - "'(, Ya-l Ya,a-l - Ya - lja · -"'(, Ya

where

O<Q<l

y~ is the composite estimate for month a,

Y,' is the regUlar ratio estimate based on the entirea

(2.6 )

\.

sample for month aj

Y~,a-l is the regular ratio estimate for month a but based

on the returns from the segments which are included in

the sample for both months a ~d a-I,

Y~-l,a is the regular ratio estimate for month (a-I) but

made from the returns from the segments which are in-

eluded in the sample in both month a and a-I.

10

The composite estimate takes advantage of accumulated information

from earlier samples as well as the information from the current one

and results in smaller variances of the current estimate and the esti

mate of the change of most of the characteristics of interest. But

the larger gains through the reduction of variances of the estimates

are usually realized for the estimate of the change. However, under

such a sampling scheme, only the within first-stage.unit component of

variance of the estimates is improved while the between first-stage

unit variance still remains the same as in the regular estimate.

This is because the same sample of first-stage units is used in every

month.

Onate [11], in developing multistage sampling designs for the

Philippine Statistical Survey of Households, adopted the ~ame princi

ple. He proposed the division of each sample barrio (a second-stage

unit that corresponds somewhat to a township in the united States)

into a small number of segments (less than 10), and imposed a

specific rotation scheme or a scheme of partial replacement to these

segments. This sampling technique was mainly intended to reduce the

response resistance of panel households. Moreover, Onate developed a

finite population theory for the composite estimator defined in (2.6)

for his specific sampling design. Rao and Graham (14] further extended

Onate's finite population theory for the composite estimator to a more

general pattern. The results presented by them were for uni-stage

sampling.

11

Other than those mentioned above, there were a few other a.uthors

who applied the partial replacement of sampling units technique to

their sampling work, for example, Ware and Gunia [18] presented a

theory which is applicable to continuous forest inventory sampling.

They also considered the problem of optimum replacement when differen

tial costs are taken into account. The method of sampling used by

them, however, was confined to uni-sta.ge simple random sampling for

two successive occasions only.

As mentioned earlier, Tikkiwal [17] in an unpublished (but

abstracted paper) proposed the technique of partial replacement of

first-stage units, but in a theoretical context different from that

treated in this thesis, as will appear in sequel.

Des Raj [3] recently proposed the selection of clusters with prob

abilities proportional to size for sampling over two successive occa

sions and indicated the application of the theory to double sampling.

From the review of literature, it is seen that the investigation

of a theory of multi-stage sampling with unequal selection probabilities

for first-stage units, and incorporating also the partial replacement

of a subset of the first-stage units on each occasion, remains an open

problem.

.. I

12

3 • A PROPOSED SAMPLING DESIGN

3.1. Description of a Method of Partial Replacementof First-Stage Units

Consider the situation when we plan to conduct successive sample

surveys for p occasions. The population U consists of N definable

first-stage units:

t ~, u2

, ~, •••••••••••••••••••••••••••••••• , ~ }

In each ~J there are Ni second-stage units (i=1,2, ••• ,N). Our objec

tive is to use n first-stage units on each occasion and selecting m

second-stage un1ts for interview. If the technique of partial replace:

ment of units is to be incorporated into the above sampling design on

the philosophy that on any two successive occasions, the first-stage

samples each of size n will contain a certain number of common units

which will be available for measuring the change over time (if ariy)

of the population characteristic of interest, the following scheme

of partial replacement of first-stage units can be used:

1. Assuming that the desired proportion of first-stage units to

be replaced after each occasion is IJ., (0 < IJ. < 1), draw a preliminary

first-stage sample of size n + (p-l)lJ.n with replacement from U. (The

probability system for selecting the first-stage units in this process

Will be defined later in this chapter.) Record the order or occurrence

of each unit. Conceivably, we would expect some identical :un:tts in the

preliminary sample since they are drawn with replacement.

2. The first-stage units which occurred from order 1 to nconsti-

tute the sample for the first occasion. Reject the first IJ.n units and

_i

13

retain the next (l-~)n units (as determined by the order of occurrence

of these units) for the second occasion, supplementing those retained

units by the next set of IJ.n units which occurred from order (n+l) to

(n+!J.n). Thus, the required s~le size of n first-stage units is

maintained on the second occasion With the assurance of having (l-~)n

units matched with those of the first occasion. On the third occasion,

reject the next IJ.n units which occurred from or~er (IJ.n+l) to 2~n while

the other (l-lJ.)n units which occurred from order 2~n+l to n+!J.n are

retained. Supplement those retained units by ~n units which occurred

from order n+J,.Ln+l to n+'2lJ.n. Do in a similar fashion for the succeeding

occasions. On the pth occasion, there will be (l-~)n units which are

matched with the (p_l)th occasion plus IJ.n unmatched units which

occurred from order n+(p-2)~n+l to n+(p-l)~n.

Example:

u =t~, ~, ~, 00. 00 • 00 •• , u60 ~

1N = 60, n = 9, p = 4 and IJ. = 3 ·

We have IJ.n = ~ =3, and n + (p-l)lJ.n = 9 + (4-1)3 = 18.

We need to draw a preliminary sample of size 18 from the above

population with replacement and record the order of their appearances.

There will be 6018 possible samples, and, of course, not all of them

are distinct. More generally, there are ~ possible samples. In our

particular example, a possible preliminary sample might be:

14

1 2 U§, 4 u5 6ui4U,2' ~7' ~4' 9

, u42 '~- 8 9 10 11 12 13 14

~3 ' U28 ' u4 ' ~5' u48 ' u19 ,~O

15 16 17 18u4 ' u4l ' u9 ' ~

,

where the superscripts indicate the order of appearance of the unit.

The structure of the preliminary sample and its four const;l.tuent

samples is sun:rma.r.ized in Table 3.1- which speaks for itself.

;5.2. Advantages of the Proposed Scheme of Partial Replacementof First-Stage units

This proposed method of partial replacement of the first-stage

sampling units when the sampling is done in two or more stages, can be

expected to minimize response resistance and other undesirable features

resulting from interviewing the same respondents over and over again.

Although the drawing of the preliminary sample of n + (p-1)lJ.n first-

stage units with replacement does permit the same first-stage units to

appear more than once, the second..stage units (and units at other

succeeding stages if more than two-stage sam.pling is used) can be

expected to be different if they are drawn independently in each re

peated first-stage unit. In sequel we shall discuss three different

methods of drawing second-stage units. Hence, the attempt to spread

the burden of reporting among more.-1!espondents is taken care of.

Another advantage of the proposed partial replacement of first-

stage units is that when these units are certain kinds of big-administra-

tive units such as villages or towns, the problem of encountering

missing units when successive sampling is done, is minimized.

.. r ,. t ... A

Table 3.1. Scheme of sample selection, partial replacement and constituent samples

Order of 1 2 3 4 5 6 7 8 9 10 11 12 13 14. 15 16 17 18appearance

~t drawn ~2 '\7 ~ '\4 u9 ~ '\4 u23 u28 ~ l)5 ~8 '\9 ~O ~ ~l u9 ~

~ample forOccasion --------------r---------------r--------------I .

1 I : r

l)2 '\7 u,: '\4 ~ ~ : '\4 ~3 ~8 \

2--------------r---------------~--------------t--------------:

: u14 ~ ~ l '\4 ~3 '\?8 ~'\ ~5 u48 IL_______________~--------------+--------------:.,..- - - - - - - --r

3 I I I

I ~4 ~3 '\?8 Iu4 ~5 '\8: ~9 '\?O u4 :L--------------t--------------i-·--------------r -------------_.

, f I4 I f1'\ ~5 '\8J ~9 ~O u4 I u4l ~ ~____________________________ L______________

t:

16

3.3. Specification of Probab1.lity System and the Methodof Selecting Sub-Units

To develop a complete multi-stage sampling design which can be

put into operation, the specification of the probability system to be

used in selecting units at each stage of sampling must be made. So

far, we have only outlined the method of partial replacement of first-

stage sampling units without specifying the probability system to be

used in selecting such units.

It is well known that in multi-stage sampling d~signs, the use of

unequal probabilities in selecting first-stage units often leads to

more efficient estimates than the use of equal probabilities.

The first-stage units may be selected with or without replacement

after each draw. However, as already stated, in this thesis the first-

stage units are selected with replacement after each draw because

selection of first-stage units with replacement confers statistical

independence between the units involved, resulting in extensive

simplifications in the estimation of variances and covariances in-

volved in the total variance of the estimator used, the expressions for

which are, to say the least, very long.

In actual applications, these selection probabilities may be

assigned proportional to the sizes of the first~stage units, ~.a., as

measured by the number of second-stage units in each first-stage unit.

(If the first-stage units are villages or towns, the size of these

villages or towns may be measured by the number of households in those

villages or towns.)

17

Next, we will consider appropriate methods of selecting second-

stage units in each selected first-stage unit. For this, a brief re-

view of three well-known methods for selecting second-stage units,

when the first-stage units are selected with replacement, will be

given.

Method I is generally attributed to Sukhatme [15]. In this method,th .

if the i first-stage unit is selected A.i

times, then mA.i

second-stage

units are selected with equal probabilities and without replacement

from that first-stage unit.

Method II is due to Cochran [1]. thIn this method, if the i

I

first-stage unit is selected A.i times, A.i

sUb-samples of size m are

independently drawn with equal probabilities and without replacement

thfrom the i first-stage unit, each sub-sample being replaced after it

is drawn.

Metbod III is due to Hartley and Des Raj [2] as pointed out by

Rao [13]. thIn this method, when the i first-stage unit is selected

A.i

times, a fixed size of thm second-stage units are drawn from the i

first-stage unit with equal probabilities and without replacement, and

ththe estimate from the i first-stage unit is weighted by A.i

•

Comments on the three methods:

Method I: In this method, it is assumed that

(i=1,2, •.• ,N).

,....

That is, the total size of each first-stage unit in the population is

relatively large as compared to the sample size at the second stage

of sampling so that the unfavorable case of drawing the same

\

18

first-stage unit up to A.i

times where mA.i > Ni has an extremely small

chance of occurring.l

In practice, especially in a large-scale sample survey, the above

problem may not arise for two reasons namely:

(i) The number of first-stage units in the population is usually

large so that the chance of drawing the same first-stage unit A.i times

where A.i is curiously large is very small.

(ii) The size of the first-stage units (namely the number of

second-stage units in each first-stage unit in the population) are

usually (or can be made) sufficiently large.

Hence, for the purpose of spreading the burden of reporting among

respondents, this method of selecting second-stage units should fit in

With our sampling design. However, when the above method was intro..

dueed, it was intended to be used only in a survey which was conducted

on one particular occasion. Hence, to apply this method is a successive

sampling scheme such as proposed earlier, some adjustment needs to be

made. In our sampling scheme, we have selected the n + (p-l)~.m first-

stage units with unequal probabilities and with replacement and the

scheme of partial replacement of units is based on these units as

described in detail previously. In that replacement scheme, it is

evident that the same first-stage unit may be selected more than once.

Hence, if we modify the assUIlI.Ption given originally to be such that the

sizes of the first-stage units in the population are sufficiently

large so that

(t = 1,2, ••• ,p)

\

t

19thwhere Ait = the number of times that the i first-stage unit is

th Iselected on the t occasion, then Sukhatme s method of selecting

second-stage units can be used. In practice, the above assumption

should not be unrealistic especially in most large scale sample

surveys.

Method II: This method is apparently free from any assumption

about the size of the first-stage units in the population since the

procedure is to select independent sets of m second-stage units with

equal probabilities and without replacement -every time from the same

first-stage unitwhich occurs more than once. We recall that in this

method, each sub-sample is replaced after it is drawn to cOIqp1ete one

particular set of m second-stage units. However, we may expect the

same second-stage units to occur more than once on a particular occa-

sion. This seemingly unfavorable event should be accepted by the

sampler since it will not increase any operational problem. All we

need to do is to go out and interview that unit once and record the

information, keeping .. the number of repetitions of occurrence as the

frequency or weight to be used in estimation procedure for that partic-

ular occasion. Should the same second-stage units be selected on the

next or succeeding occasions due to the nature of our sampling design,

the sampler also should accept these (unfavorable) events and go ahead

to interview those units according to the number of occasions in which

they are included in the sampling plan. These events should not be

many, and in View of maintaining the rate of response, we may instruct

the interviewers to try their best to explain to such would-be

respondents as to why they are interviewed for several occasions. In

20

conclusion, this method of selecting second-stage units will be adopted

in our partial replacement scheme in preference to Method I where the

assumption

PNi 2: m ( I: ;\'it)

t=l

may not be satisfied.

Method III: This method does not seem to fit into our sampling

design by its very nature, !.~., it will not help in spreading the

burden of reporting among respondents. Hence, we will not consider it.

.'

. 21

4. ESTDvlATION THEORY

4.1. Introductory Remarks

As stated earlier the sampling theory shows not o~ythe sampling

design, but also shows the estimation theory which follows "from it.

We will not consider the estimation procedures which can be made under

the sampling theory to be proposed. It should also be emphasized that

the estimation theory in statistical sampling generally does not de-

pend upon the concept of distribution of random variables. However,

the estimation theory in Statistical Sampling partially relies on basic

criteria in the classical estimation theory such as unbiasedness,

minim.um varianc,)../ or m.inimum mean squar~/, which are known to char

acterize a good estimator. When the two properties are about the, same,

. IIwe may have to add another criteria which by common sense is ease of

computation" so that only one estimator stands out as the best choice.

However, the added criteria has become less significant in the coun-

tries where modern electronic computers are extensively employed.

To develop an estimation theory which will be applicable to the

proposed sampling design, some of the criteria mentioned will be

utilized. The essential notation to be used is as follows:

2./However, Godambe [5] and more recently Kool> [9] have demonstratedthe non-existence of minimum variance unbiased estimators when unitsare drawn with unequal probabilities •

.!t/In the case of an unbiased estimator, these two are the same

since Mean Square Error = Variance + (Bias)2.

•

22

th th= the variate value of the j second-stage unit in the i

thfirst-stage unit on the t occasion.

tT = the total of population characteristic of interest on

ththe t occasion (t = 1,2, ••• ,p).

N = the number of first-stage units in the population.

n = the number of first-stage units in the sample (Le., the

first-stage sample size on each occasion).

Ni the number of second-stage units in the i th first-st~e

unit (i =1,2, ••• ,N).

m = the number of the second-stage units in the sample

(chosen to be equal for every ~irst-stage unit, to fit

with our sampling design) •

Ni . th= ~ t Yi ., the population total for the i first-stage

j=l J thunit on the t occasion.

Pi > 0, (i =1,2, ••• ,N), the probability of selecting the i th

Nfirst-stage unit such that, ~ Pi =1.

i=lS is a set of first-stage units selected in a specified order

and is a subset of the preliminary sample of n + (p-l )lJ.n

first-stage units. The size of this set (sample) will be

clear from the context in which it is used.

Other symbols will be defined where they are used.

In this thesis, we will be concerned in estimating the total of

the population characteristic and the change or difference between the

totals of population characteristics on two successive occasions. If

interest lies in the~ of the population characteristic and the

..

•

change or difference between the means of population characteristic,

the theory can also be used 'With a little change entirely in the

multiplying constant{s) that enter into the expressions for the esti-

mators and their variances.

For the development of the estimation theory, we will adopt the

method of selecting second-stage units as suggested by Cochran [1].

Although Sukhatme's method [15] of selecting second-stage units is not

so complex in principle and can be adopted, it is felt that such a

method requires a restrictive assumption about the size of the first-

stage units and leads to a complicated expression for the variance.

Thus, in the development of the estimation theory that 'Will follow,

whenever a first-stage unit is selected, m second-stage units are

independently selected from that first-stage unit 'With equal ~rob-

abilities and without replacement.

4.2. Estimation of Total for the First Occasion

To estimate the total of the population characteristic for the

first occasion under the proposed sampling design, is a straight-

forward procedure. As a background for comparison with the estimation

procedure for the second occasion, and as other derivations of vari-

ances and covariances have a bearing on these results, it is worthwhile

to include the relevant theory here, despite the fact that it is

already well known.

N NiTo estimate 1T = ~ ~ 1Yij' !.~., the total of the char

i=l j=l

acteristic of interest of the population on the first occasion.

24

An unbiased estimator of IT

\"- 1 n NiT = - .E -

1 n i=l Pi

m.E

j=l(4.1)

We show that IT as given in (4.1) is unbiased. Since two~stage

"-sampling is used, the expected value of IT is easily obtained by

applying the well-known theorem on conditional expectation; n~ely

(4.2)

•

..

where in our context E(·I S) refers to the conditional expectation

given the first-stage sample.

We now apply (4.2) to find

E( T) = E[E ! ~ Ni ~ lYij Is]1 n i=l Pi j=l m

Now

so that after some simplification

whereNi

lYi = j~ lYij •

25

FuI;.ther

\ N= . L: 1Yi = 1T

J.=l(4.4)

So that

,..Hence, IT is an unbiased estimator of IT as claimed.

,..The variance of 1T can be derived by several methods. One m..ethod

which seems to be most convenient for multi-stage sampling is t9 apply

an important theorem on conditional variance formalized by Madow (1949);

namely

We apply (4.5) to find the expression for Var (IT) .

Consider first, Var ~E(lTIS)}. We have

E( Tis) =1 ~ E(Ni ~ lYij ·Is) 0

1 n i=l Pi j=l m

(4.6)

..1= -n2

n

since

= 1 ~ P (1Yi _. T)2n i=l i Pi 1

E (lYi ) T for all i as demonstrated at (4.4).Pi = 1· ,

\

26

Now, con,si<ier the second part of the variance.in (4.5); nameiy,

E Var (lT1s) • Returning to (4.1), we have

(4.8)

It is well known that

where

(Ni

-m)

(Ni-1)

and

1~ ,

so that

and hence

Var ...L2n

\

N Ni 22 (N -m)

E {var (lTla)1 n 10"i i::; T .E Pi (-) - (Ni -1)Pi mn i=l

1 N ~ 1~ (Ni-m)= .En i=l Pi m N -1i

'"Therefore, the expression for Var (IT) is

(4.10 )

..::: 1. [~ p

n i:::1 i

27

(Ni-m) ](N.-1)

J.

(4.11)

::: ~ [the between first-stage units variance + the

average within first-stage units variance].

4.3. Estimation of Total for the Second Occasion

On the second occasion, we wish to estimate

NL:

i=l

We recall that the sample for this occasion is made of the first-stage

units appearing from the (IJ.n+1)th to (n+!J.n)th draw. Of this set of n

first· stage units, those units appearing from the (IJ.n+1)th to the nth

draw constitute the matched part. Those units -appearing from the

)th th(n+1 to the .(n+!J.n) draw constitute the part which replaced the IJ.n

units dropped after the first occasion.

If there is 'a change over time of the characteristics of the

population, then the (l-lJ.)n first-stage units which are kept in common

between the two occasions should serve as a natural measure of such

change. We will consider several possible estimators which can be

used to estimate the total for the second occasion. These estimators

are linear combinations of estimates based on different parts of the

sample and take advantage of past as well as present information from

the sample to provide an estimate for the present occasion (in this

case the second occasion).

28

4.3.1. A Linear Composite Estimator

We represent the structure of our sampling procedure covering the

first two occasions diagrammatically as follows:

1st occasion

IJ.n units (l-lJ.)n units

I t 1\

2nd occasion ! I(l-lJ.)n units IJ.n units

The set of units connected by the two way arrow represents the

matched portion. The linear co~osite estimator for the second occa-

sion is

• (4.12 )

where *2T = the linear composite estimator of the total for the

second Hoccasion:

'"1T = the unbiased estimator of the total for the first

occasion as defined in (4.1)

= 1 ~ ~ ~ 2Yij E

(l-lJ.)n i=fJ.n+1 Pi j=l m

(4.14)

'2./ 29A 1

n+J,J.n Ni m 2Yij2T = - Z Zn i=j..l.n+l Pi j=l m

!.!:.., the unbiased estimator of 2T, based 0 n all of the n first

stage units for the second occasion.

Remarks:

(i) The subscript i in the summations of (4.13), (4.14), (4.15)

refers to the order of occurrence of the relevant first-stage units

which are recorded in the preliminary drawing procedure.

(ii) Although lYij and 2Yij, are not the same, the associated

probabilities Pi are conservative.

1'". , /,'to.

Expected Value Of 2T. It is easy to verify that' 2T'. is

• also unbiased. We have

It can be shown along the same lines as in (4.4) that

Hence

'2./This estimator namely /r would be used if the sampling was

carried out only on~ occasion, or if past information from thefirst occasion is totally ignored.

4.3.1.2.~' ~

Variance of 2T. We recall that

30

...The expression for Vax (IT) has been established in (4.11),

namely:

(4.16)

The expressions for other relevant variances in (4.16) can be...

derived in the same fashion as for Vax (1T) • They are:

(Ni-m) ](Ni-1)

(4.17)

... 1 [. N 2Yi 2 N ~ 20";Vax ( T) = - Z P (- - T) + Z - (-)

2 n i=l i Pi 2 i=l Pi m

,

[

N lYi 2 N ~ 10"; (N1-m)]. Z Pi (7 - 1T) + Z P (m) (Il -1)1=1 1 1=1 1 1

( 4.18)

(Ni-m) ](Ni -1) • .

(4.19)

,31

where in (4.17) and (4.19)

and

= ~i 2Yijj=l Ni

The expressions for the relevant covariances in (4.16) will nowA A

be derived. Consider first, Cov (2,lT, 1,2T).

From the structure of the two estimators we have explicitly:

A

A (1 n Ni m 2YijCov (2,lT, 1,2T) = Cov (l-~)n L: L:

Pi m ,i=~n+l j=l..

1n Ni m

l~iJ J(l-~)nL:

PiL:

i=~n+l j=l

To obtain the expreSsion for the above covariance, we apply the

theorem on conditional covariances formalized by Madow [1 0] which is

as follows.

If U and V are random variables and A is a random event, then

Cov (U, V) = E[Cov (UIA, viA)] + Cov [E(uIA), E(V!A)]

A A-

To apply the above theorem for finding Cov (2,lT, 1,2T), letA A

2,lT play the role of U, 1,2T play the role of V, and S, here the set

of first-stage units selected from (~n+l)th to nth draw, play the role

of A. Then, it is not difficult to show that

~ 2Y

ii=jJ.n+l Pi

Ni, where 2Y' =!: 2Yi'

J. j=l J

and where

t1 n

COY !:. ( I-jJ. )n i=fJn+l

..

This leads to

( )

_ 1 N 2Yi 1YiCOy E(2,lTIS), E(1,2T1s) - (l-fJ)n !: P. (- - T)(- - T).

1=1 J. Pi 2 Pi 1

~ (Ni )2 ~i (2Yij - 2Yi)(lYi {lYi)

i=jJ.n+l Pi j=l Ni

1 (Ni-m)iii (N

i-1)

since second-stage units are selected without replacement in each

selected first-stage unit.

33

Hence

(4.21)

Corribining (4.20) and (4.21), we thus obtain

+ ~ ~ ~i (2Yij - 2Yi)( lYij - lYi)

i=l Pi j=l Ni(4.22 )

We may interpret the result given in (4.22) as follows.

The total covariance is made up of two parts name:Ly, the between

first-stage units covariance and the average within first-stage units

covariance.

We also notice that in our context, the above covariance virtually

measures the auto-regressive·· nature over time of the characteristic

under- .study.

Next, we consider cov-(lT, 1,2T). To establish the expression for

this covariance, we proceed as follows.

•

J

A A

From the structure of IT and 1,2T, we have

This can be rewritten as

A 1 IJ.n Ni mCOy (IT, 1,2T) = COY - (E -- E

n i=l Pi j=1

1 ~ Ni ~(l-';)n P

r- i=J.l.n+l i j=l

= 1 [ IJ.n(1-) -COY ( En IJ. n i=1

By virtue of mutual statistical independence between the first-

stage units u1 ' u2 ' ••• , uJ.l.n and ulJ.n+l' uJ.l.n-t2' •••••••• un' it can be

shown that

IJ.n N mCOY ( E pi E

i=l i j=l, (4.24 )

35

The second expression in the square bracket, namely

~, n Nim lYij n Ni m lYij

Cov( .E .E .E .E )Pi

,Pii=lln+l j=l m i=lln+l j=l m

is readily recognized as

n Ni m lYijVax ( .E .E )

i=jJ.n+l Pi j=l m

"which can bederived in the same way as Vax (1T), and we obtain

n Ni

mCOy (.E -.E

i=J.1n+l Pi j=l

lYijm '

Combining these results, we thus obtain

(Ni-m)}(Ni-l) •

(4.25 )

(4.26)

f

,.The very nature of COy (l~\ 1,2T) as revealed by (4.26) shows

that it is quite different from COy (2,lT, 1,2T).'-'

The expressions for other covariances involved in Vax (2~) will

now be derived along the same lines. By definition,

.,

tl {lJ.n Ni m lYij n Ni m lYij }= Cov.- ~ -- ~ + ~ -- ~ ,n i=l Pi j=l m i=lJ.n+l Pi j=l m

Using the same procedure as in the development of the expression

'" ....for COy (1,2T, 2,lT), it will be found that the second part, namely

~ nNi m lYij n Ni m

2:1JJCOy ~

Pil: , ~

Pi~

i=j..m+l j=l m i=lJ.n+l j=l

(4.28)

37

Now consider

r~nN

im lYij m N

im

2:iJ JCov L:Pi

L: , L:Pi

L:i=l j=l m i9J,n+l j=l

Iln n iNi m lYij N.e ~ 2mY.eJ] 6/

= L: L: Cov pL:. , P "i=l .e=lln+l i j=l m .e j=l

Again applying Madow's theorem, we find

·\Ni m lYij N.e ~ 2Ym.eJ}Cov P L:, , P "

i J=l m .e j=l

= Cov rE (:i ~ lYij I 8) ,L i j=l m

..'

( Ni

mlYiJ 18 N.e m 2Y.ej

I SJ .+ E Cov (Pi L: P.eL:

j=l m ' j=l m

The first part is

t Ni m lYij N m

2Y.ej I 5)Cov E(Pi

L: I 8 ), E(..1. .Ej=l m P.e j=l m

tlYi 2YJ} N N lY' 2T,e

= Cov Pi ' = .E L: Pi P (~ - IT)(p- - 2T)P.e i=l .e=l .e Pi ,e

2/It may be noted that the derivation here is diff~rent from thatn Ni m lY1J n N1 m 2Yij

of COV[ L: p ~ m ' L: P I: m ]; this is due to thei=lln+l 1 j=l 19J,n+l i j=l.

fact that the latter is the planned matched portion. The relevant probabilities are quite different; this should be apparent when one recallsthat the index i in the summation signs refers to the order of theoccurrence of the first-stage units. -----

38

For the second part, consider the covariance expression under the

expectation sign, we find, as in the derivation leading to (4.25) that

[Ni m lYij N,e m 2Y

m,ej I S)~E Cov (- !: . Is, -!: .

Pi j=l m P,e j=l

= 0

and hence

Cov {~n Ni ~i=l Pi j=l

~ 2~ijJ =j=l

o •

Similarly,

A A [1 n Ni m 1Yij 1 nT)J.n Ni ~ 2Yij]Cov (1T, T) = Cov - I: - I: , - I: - L..

2 n i=l Pi j=l m n i9.ln+1 Pi j=l m .

39

1Yijm ),

n Ni m 2Yij n+i-!n Ni ~ 2Ymij)J(I: pI: m + I: P L..

i9.ln+1 i j=l i=n+1 i j=l

_ .1:. [- tJ..Ln Ni m 1Yij- 2 -Cov I: P I: m '

n 1=1 i j=l

t n Ni m 1Yij n Ni m 2YiJ... ]+ Cov . I: - I: , I: - I:

i=J..Ln+1 Pi j=l m 1=J..Ln+1 Pi j=l m -

\ ~nNi m 1Yij n+i-!n Ni

m2~iJJ+ Cov I:

~I: , I:

PiI:

i=l j=l m i=n+1 j=l

{n N m 1Yij n+i-!n Ni m27r J]+ Cov I: pi I: I: I: J •

m,

Pi j=l mi=J..Ln+1 i j=l i=n+1

The ~, third a.nd fourth covaria.nces can be shown to be zero,

and we are left with

1 [ (n Ni= - Cov I: -n2 i=J..Ln+1 Pi

mI:

j=l

lYijm '

..'

~ 2:ij J]j=l

~J

40

Apart from constants, we find as in the derivation leading to (4.29)

that

(Ni-m)l

(Ni-l)}

(4.30)

'" '"The same procedure when applied to Cov (2,lT, 2T) and

'" '"Cov (1,2T, 2T), yields

and

2 N lYi 2By writing leTby for E Pi (p - 1T) ,

i=l i

N ~2

(Ni-m)2 l eTileTwy for E

Pi (Ni-1) ,

i=lm

(4.32)

.' 41

(Ni-m)(N

i-1) ,

and

we arrive at more concise expressions involved in the variance function

given by (4.16), viz.

COy (T T) = (l-~)n1 ' 2 2n

In terms of the notation given by (4.33) we find from (4.16)

Var (2~) - Q2 [1 (0:2 + 0"2 ) + 1 (0:2 +0"2 )- n 1 by 1 wy (1-1l )n 2 by 2 wy

42

1 2 2 2+ "'(l:---=Il~)-n (lO"by + 10"WY) + Ii (1.20"byy + 1.20"wyy)

_ g (0:2 + 0"2 )... 2 . ( 0: + 0" ~2 1 by 1 wy (l-ll)n 1.2 byy 1.2 WYYJ

+ (1_Q)2 [1 (~ + 0"2 )] + 2Q(1-Q) [(l-ll)n ( 0:n 2 by 2 wy n2 1.2 byy

+ 0" ) + 1 (0:2 + 0"2 ) _ 1. ( 0: + 0" )].1.2 wyy n 2 by 2 wy n ,1.2 byy 1.2 wyy

(4.34 )~

4.3.1.3. Note on the Estimation of Var (2~)' The expression forA.

Var (2T) involves the constants Q, IJ. and n and the population

2222+values of 10"by + 10"WY' 20"by + 20"WY and 1.20"byy 1.20"wyy· An un-

Abiased estimate of Var (2T) can be made by substii:;uting unbiased

. 222 2estl.mates of 10"by + 10"WY' 20"by + 20"WY and 1.20"byy + 1.20"wyy

respectively in (4.34). By virtue of our sampling design, it can be

shown that

2 2 n 1Y~ A. 2(i) 10"by + 10"WY is unbiasedly estimated by ~ (p - 1T) / (n-1),

i=l i

where

A _

2T and 2Yi are as defined previously.

...

The verification is given in the Appendix •

4.3.1.4. *Efficiency of 2T.

43

To see the usefulness of the linear

and

1Vy for 2 210"by + 10"WY ,

2Vy for 2 220"by + 20"WY ,

V for ~ + 0" ,1.2 Y 1.2 byy 1.2 wyy

to obtain more compact expressions for variances.A

The form of 2T has been given previously at (4.15), namely

A = 1 n~n Ni~T ~ Pc;. n i=lln+1 i

mr:

j=l

with variance given in (4.33), namely

= 1. (0".2 + ,; ) =n 2 by 2 wy

Using the above symbols in (4.34), and after some algebraic

manipulation, we find

1/This estimator would be used if no attempt is made to utilizethe prior information from the first occasion or if the sampling isconducted only for one occasion.

44

tVar

...

Var (2~) = Var (2T) + l~) [1} + 2:1- 2Q;t ~ + (l~J l.~Vy(4.35)

To make a meaningful comparison between the efficiency of 2~ and

"2T, we will make the following assumptions:

(i) V = V = V ~/ (say)1 y 2 Y Y

V(ii) p = 1.2 y

-V;V;~

lYi and 2Y

ip is the correlation betweenPi Pi

Under the above assumptions, (4.35) yields the relation

~ A 2Q2 V [Q 1 VVar (2T) = Var (2T) + (l~~) -! -2Q~ .1 + (l~~)J p -!' .

~/ThiS assumption is referred to, in time series analysis, assecond order stationarity or weak sense stationarity.

.,

tJ

~When p = 0, the expression for the variance of 2T is

This implies that in such cases (p=O), the linear composite

estimator will be less efficient than the regular unbiased estimatorA

2T. This should be no surprise to the sampler because in such cases

(i.e., no correlation over. time), we would not spend any effort using

the prior information from the first occasion in estimating the

characteristic of interest for the second occasion.

In actual application, we would expect that ~he correlation p is

sufficiently high so that estimation by a composite estimator is worth-

while. For this, we will now examine the nature of the gain in effi

ciency of 2~ over /r for some realistic values of p}.1 Since the~

nature of the gain in efficiency using 2T, also depends on the wei¢ht

Q and the proportion of first-stage units replaced namely IJ., we

consider such gain for some sets of values of Q and IJ. which ~ be of

. . 101practical interest.---

21The estimate of p can be made from the samples. The formula isI I

n {'i A- 2Yi A-

I: (p- - 1 2T) (p- - 2 IT)= i=pn+l i' i'

p II ~ (lY~ - T)2 ~ (2Yi _ T)2]i=jJ.n+l Pi 1,2 i=pn+l Pi 2,1

~IThe optimum value of both Q and IJ. will be discussed later.

Case I:

Here

~ AHence, it will be seen that 2T will be more efficient than 2T

if [~ - if. p] < 0 ,

4!.~., if p > "7 <= ·57 .

The per cent gain in efficiency of 2~ over 2T as measured by

46

is

The gain as a function of p is tabulated for some values of p

in Table 4.1.

Case II:

Here

Var <2~) =Var <i) + [it -~ p]:t ·We see again that 2~ will be more efficient than 2T if

!.~., if p > ~ = .60.

j

...

47A A

The percent gain in efficiency of 2T over 2T in this case is

( ) _ (5p-3)G2 P - l5-5p x 100.

The gain is also tabulated in Table 4.1.

Case III:

Here

(2~)A [~- t p] V

Var = Var (2T) ... .Ln ,

~ A

and thus 2T is more efficient than 2T if

!.~., if P > .67 .~ A

The percent gain in efficiency of 2T over 2T in this case is

The nature of the gain for some values of p is also tabulated in

Table 4.1.

4.3.1.5. Choice of Q in the Linear Com,posite Estimator. The

lit "gains in efficiency of 2T over 2T considered previously are for

1some specific sets of values of Q and IJ.. The weight Q =2' is

selected because it appears to be the most natural choice to begin

with. The replacement rates of' IJ. = t ' ~, and ~ are the most

natural to choose and perhaps will be of' practical interest.

,. ,.Table 4.1. Percent gain in efficieney of 2~ over 2T

.- ;

Case I: p~/=.6 p=·7 p=.8 p=·9 p=·95

1Q=-2< 1.00 3·90 7.14 10.60 12.41

1f.L = '4

Case 2: p=.6 p=.7 p=.8 p=·9 p=·95

1Q=-20.00 4.35 9·09 14.29 17·07

. 1f.L =-

3

Case:5 : p=.6 p=.7 p=.8 p=·9 1'=.95

1Q=-2negative 2.56 11.11 21.21 26.98

1f.L=-2

!:/Note: the number under each value of p is thecorresponding percent gain in efficiency.

48

~

From the expression for Var (2T),we will see that apart from the* ,.

value of p, the nature of the gain in efficiency of 2T over 2.T,

generally depends on both the values of Q and fJ.. In practice, the

sampler may not have enough freedom to choose the value of fJ. and may

have to choose the one that fits best with practical conditions of the

problem such as the costs ,of the survey, etc. Under suchconditions,

it becomes necessary to determine the best choice of the weight Q to

be used in the composi"tie estimator (4.12.) so that maximum gain in

efficiency (for a fixed fJ.) is realized.

We will now proceed to determine the optimum value of Q for some

specific replacement rates (!.!.,

practical interest, and the gains

realized.

Case I:

1fJ. = 4'

for some values of fJ.), 'Which are ·of* ,.

in efficiency of 2.T over 2.T therein

ItThe variance of 2.T given in (4.36) can be rewritten as

Var (2~J • [1 + 2Q2~ + (2Q1..e -(i~)- 2QVJ p] 1When fJ. =~, the above expression for Vax (2') simplifies to

[ 1 +~ - (~ +~ Jp] 1·For fixed n and V I the optimum va1ue of Q is given by

[

2 ,Y 2]V~ :1 + 2§ - (~ + f)p -f = 0

50

!.~.,

vJ. = 0n '

and we find

Q - -.2E- 8-2p

Table 4.2 shows the optimum values of Q and the percent gain in

efficiency in this case for some values of p , which are likely to

occur in practice.

Table 4.2.

p

.6

·7

.8

·9

·95

1Optimum value of Q for ~ =4 and percent gain in efficiency~ ,..

of 2T over 2T

Optimum value of Q!! %Gain in efficiency

.26 4.14

.32 5·90

037 8.11

.44 10.85

.47 12.45

!/Rounded to the nearest two digits.

Case II:

In this case, (4.37) simplifies to

[1 + '12- (23'1 +~) pJ ~ .

~

Minimizing Va:r (2~)' we find that the optimum value of Q = (3-P)

51

The optimum values of Q as function of p and the percent gain

in efficiency for this case are tabulated in Table 4.;.

Table 4.;. 1Optimum value o~ Q for ~ =3 and percent gain in

efficiency of 2T over 2T

p Optimum value of Q %Gain in efficiency

.6 .25 5.26

·7 .;0 7.6;

.8 .;6 10.74

.9 .4; 14·75

.95 .46 17·14

Case III:

1IJ. =-:2

In this case, (4.;7) reduces to

[1 + 2Q2 _ (Q + Q2 lP] ~ ,*Minimizing Var (:2T) results in the optimum value of Q = 4-~p •

The optimum values of Q as a function p and the percent gain in

efficiency for this case, are tabulated in Table 4.4 •

..2e.... 1We see that the optimum values of Q which are 8-2p for IJ. =4 '

;~p for IJ. =~ , and 4-~p for IJ. =~ approach ~ as p tends to

unity.

J..

Table 4.4. 1optimum value of Q for ~ =2 and percent gain inA A

efficiency of 2T over 2T

52

p optimum value of Q %Gain in efficiency

.6 .21 6.87

·7 .27 10.40

.8 ·33 15·33

·9 .41 22·56

·95 .45 27·23

1Hence, in practice, the choice of Q =2 should be satisfactory

when the correlation p is high, meaning any value between .9 and 1.

The deviation from maximum precision of the linear composite estimator

2~ will be small but considerable reduction in computational work is

achieved. However, for a low value of p, the sampler should try to

use the exact optimum value of the weight Q so that the entire

complex sampling design will be worthwhile.

4.3.1.6. Choice of ~. In the previous section, we have

considered the optimum value of Q for some specific values of ~,

1and found that for a high value of p, we may use Q =2 as the weight

in the linear composite estimator without sacrificing much efficiency

and also avoiding tedious computations. Another problem of interest

is to determine what is the best choice of ~ (the replacement rate)

when Q = ~ is used. Further, how would the gain in efficiency change

from the use of optimum value of ~ as against natural values of ~

such as ~ = ~ , ~ , ~ which have been considered previously.

.J

53

We proceed to determine the optimum value of I.l. as follows:

1Substituting Q =2' in (4.37) and after some simplification, we

have the following variance function for 2~

~2-~li:~r~ ~

Setting ~~ [(2-~Jfi:nl) ~] = 0

leads to the following quadratic equation in I.l.:

PI.l.2 - 2pI.l. + 2p - 1 = 0 •

Solving we find

= 1 + ""p(l-p)I.l. p

(4.38)

"

Since 0 < I.l. < 1, we see that the only admissible root of (4.38)

is

which is the required optimum value of I.l..

Table 4.5 shows the optimum value of I.l. as a function of p and. ~ A

the percent gain in efficiency of 2T over 2T •

.We see from Table 4.5 that when p is high, there is an

appreciable increase in the gain in efficiency. However, the optimum

value of I.l. may not be feasible when p is as high as .95 simply

because it involves replacing over three-fourths of the first-stage

1units. In such case, the choice I.l. =2' might be used without

sacrificing much efficiency.

Table 4.5. 1Optimum value of lJ. for Q =2" and the percent gain in

* "efficiency of 2T over 2T

54

p Optimum value of lJ. %Gain in efficiency

.6 .18 1.02

·7 .34 4.30

.8 ·50 11.11

·9 .66 25·00

·95 ·77 39·39

Such choice (i.e., 111 = 2" ) should serve well both from the point

'(

of view of the sampling operation and ease of computation.

4.3.1.7. Simultaneous Optimum. Values of Q and 11 in the Linear

Composite Estimator. The optimization so far considered is for Q

and 11 separately, !.!:.., the optimum. value of Q was determined by

holding 11 fixed and vice versa. In practice, one might be interested

in searching for the best combination of Q and 11 so that the linear

~composite estimator 2T as defined in (4.12) will have the smallest ..

variance or equivalently, the gain in efficiency using the linear

*composite estimator 2T will be fully realized. We will now consider

the simultaneous optimum. values of Q and 11.

We recall that

" [1 + 2Q2( 2Q2 2QIl - 2Q21l2) p] V

Var (2T) 11 + ..Jl.- (1-11 ) (1-11) n

2e C1 (Q"g.) . n ' say.

J

55

For fixed values of n and V, the simultaneous optimum valuesy

of Q and ~ maybe defined as the pair of values of Q and ~

which minimize Cl(Q,~) defined in the above identity.

Setting

we have

and

2Q2 [(l-"ll:"~~-l~ + [(1-" lt4Q)! - 2Q - ¥<2" ]

- (2QJ12

- 2QJ1 - 2Q2"2) { -1 )] (1_~)2 = 0

and noting that 0 < ~ < 1 we find, from (4.39 )

and from (4.40)

Q = (p _ 2p~ + p~2)

(1-2p~ + p~2)

From (4.41) and (4.42), we have

(4.39 )

(4.40)

(4.41)

(4.42)

... 56

This leads to a cubic equation in ~, namely

(1l-1) (P1l2

- 21l + 1) = o.

Since ~ f 1 , we have

2pll - 21l + 1 = 0,

yielding the roots

Il = (4.44)

.....

The admissible value of ~ must be

1 - .y 1-pP

(4.45 )

Substituting the value of ~ from (4.45) in (4.41) we obtain

(4.46)

The form of Q is quite suggestive in that it approaches ~ when p

approaches 1.

It can be verified that

~ = 1 - -v;;:;;p

and

,

4.3.1.8. Expression for the Minimum Attainable Variance of the~

Linear Composite Estimator 2T. If the simultaneous optimum values of

Q and ~ are actually used, from (4.37) we have

57

t·= [1 + 2a(1- -{l:;)} 2. [1 - ..-P;}

1 ... (1 ... ~),p

p

1- (1- -y:;:p)p

upon simplification, the expression is

~ [1 "'~J2J VVar ( T) = 1... - {1 - V1-p ..:Y-2 opt. 2 n

Or equivalently,

v..:Y-n

(4.48)

.... ....The percent gain in efficiency of 2'2 over 2T for the

simultaneous values of Q and IJ. as given by (4.46) and (4.45) 1s

58

The simultaneous optimum. values of Q and fJ and the percent~

gain in efficiency of 2T over 2'£' when such values are used, are

tabulated in Table 4.6.

Table 4.6.~ A

Percent gain in efficiency of 2T over 2T for

simultaneous optimum values of Q and fJ

p Optimum fJ Optimum. Q %Gain in efficiency

.6 .61 018 7·25

07 064 023 11.43

.8 069 028 18.05

.9 076 .34 30·55

095 082 .39 ·43.67

Comparing the nature of the gain in efficiency when the

simultaneous optimum. values of Q and fJ are used, With the previous

cases, we will see that there is an overall increase in the gain from

every case considered previously. When the exact optimum. values of Q

and fJ are not too convenient to be used, the sampler may adopt the

1 1choice of fJ:= 2 and Q:= 2 with a slight loss in efficiency. But

this is compensated by a simpler replacement procedure. A better

1strategy is to use fJ:= 2 (1.!.0' replace half of the first-stage

units on the second occasion) .and use the optimum. weight Q corespond-

ing to the p-value applicable, given in Table 404. This may be done

when it is evident that the correlation p is between 0.6 and 0.8.

1 1When p > 0.8, the choice of fJ =2 and Q = 2 still produces a

substantial gain in efficiency and may be adopted for simplicity.

59

4.3.2. A Modified Linear Composite Estimator

From the proposed sampling design, another estimator that bears a~

close resemblance to the linear composite estimator 2T may be

constructed. The structure of such an estimator which we Will refer

to as a modified linear composite estimator is similar to the limiting

form of Jessen's estimator [8] and to the estimator recently considered

by Des Raj [2]. It is

~

Q[lTA

1,2T] + [1 - Q]A

T' = + 2,lT - 2,2T2

where

A 1 nTJ.,Ln Ni m 2Yi2,2T =-- L;

PiL;

lln i=n+l j=l m (4.50 )

A

1,2T and Q are as~

2T' is unbiased.

!.!., the unbiased estimator of 2T, but based on those first-stage

units which are selected to replace the lln first-stage units asA A

described in the sampling design. IT, 2,lT,

defined preViously 0 It is easy to verify that~,

Intuitively, the estimator 2T will have a less complicated

expression for its variance than 2~. This is by virtue of ourA A ,A

sampling design. 2,2T is statistically independent of IT, 2,lT,A A

1,2T and hence there will be no covariance between 2,2T and each of

the three just mentioned. However, we should expect that the ~ppro-

*priate weight Q used in the construction of 2T may not be appro-~,

priate for 2T. We will consider the nature of the gain in efficiency

~Iof 2T for some specific values of II which might be of practical

interest.

• 60A

For this, we first derive the expression for the variance of 2~'.

From (4.49), we have

A

Var (2T') = Q2[var (IT) + Var (2,lT) + Var (1,2T)

A A

- Cov (1 2T, 2 2T)] •, ,

It can be shown by a method similar to and leading to (4.25) that. - ........ A ....

the last three covariances namely Cov (IT, 2,2T), Cov (2,lT, 2,2T) ,.... A

Cov (1,2T, 2,2T) are each zero, so that we find

(4·51)

Using previous notation, it can be shown that

(4.52 )

(4.52) together with (4.33) leads to the expression

~, 2rl ) 1 () 1 1Vax (2T ) = Q In (lVy + (l-~)n 2Vy + (l-~)n (lVyj

+ 2Q2[~ P-V<lVY)(2Vy) - ~ <IVy)

- (l:~)n P-V< IVy) (2Vy) J+ (1_Q)2 ~~ (2Vy)'

Under the assumption that lV = 2V = V ,(4.53) reduces toy y Y

V (~i) = [Q2~2<l-2j?) + ~(2Q-l) + (Q-lll '!.z.ax 2 ~(l-~) J n

61

~We will first consider the efficiency of 2T' as compared to the

original lineax composite estimator 2~ when the weight Q = ~ is

111used for the cases: ~ = 4" 3' and 2' respecti,vely.

Case I:

In this case, (4.54) leads to a relation

Var <2~') =[1 +~ - %pJ t= Vax ( ~) + [5 - 2pl ~

2 . 12 J n

> Vax (2T) for all 0 < P <1 •

p >

A ::=:It has already been demonstrated that Vax (2T) > Vax (2T) for

~ ~

.57. We see that in this case, 2T' is less efficient than 2TA

and also less efficient than the simple estimator 2T.

62

Case II:

In this case, (4.54) leads tO,a relation

'!.zn

\

Since we have shown that Var (2T) > Var (2~) for p >.6, we see

again that in this case 2~' is less efficient than 2~ and also lessA

efficient than the simple estimator 2T.

Case III:

In this case, (4.54) leads to a relation

Var ( ~i ) = [1 + (1 - 2p ~ ~2 4 . J n

= V ( ., TA ) + (1 - 2p) '!.zar 2 4 n •

* AHence, it is clear that Var (2T') will be less than Var (2T)

1when p > 2' .~

To see how the nature of the gain in efficiency using 2T' changes

*when it is used instead of the original linear composite estimator 2T,

* Awe compute the percent gain in efficiency of 2T' over 2T which for

.....:.,

this case is

G ( ) = (2p-l) x 1005 P 5-2p

The gain for this case is tabulated in Table 4.7 for some values of p.

I. ~, " 1Table 't.7. Percent gain in efficiency of 2'.1.' over 2T for fJ. =2' '1Q=-2

p 10 Gain in efficiency

.6 5.26

.7 11.11

.8 17.64

.9 25·00

·95 29·03

*Comparing with the gain in efficiency using 2T as given in

Table 1, we see that for each corresponding value of

higher gain in efficiency.

*T'p, 2 shows a

'"To get the idea about the full potential of 2~' when we are

free to choose the weight Q so that maximum gain in efficiency for

fJ. =~ , ~ and ~ is realized, we will now consider using the optimum Q

for each case and compare the nature of the gain in efficiency to thatA

of 2T.

Case I:

64

From (4.54)

Minimizing Var

= [(17 - 2p) Q2 _ 8Q + 41.::z .l 3 J n

~

(2T' ), with respect to Q, results in the optimum

value of Q given by

The optimum values as a function of p and the percent gain in

*efficiency of 2T' over 2T when optimum Qr S are used are- tabulated

in Table 4.8.

Table 4.8.

Case II:

~Optimum value of Q to be used in the estimator 2T for

1 A

J..l. = 4 and the percent gain in efficiency over 2TI

P Optimum Q %Gain in efficieacy

.6 .76 3·94

·7 ·77 8.33

.8 .78 13·23

·9 ·79 18.75

·95 .80 21.95

1J..l.=,

In this case~(from (4.54)),

Var (2~') = [~(5-P) - (6Q-3~ ;:

The optimum value of Q is found to be

Opt. Q := ~ •

Table 4.9 shows the nature of the optimum values of Q and the

65

AAt

gain in efficiency of 2TA

over 2T •

Table 4.9.

Case III:

*Optimum value of Q to be used in the estimator 2Tf

when ~:= ~ and the percent gain in efficiency over 2T

p Optimum Q %Gain in efficiency

.6 .68 4.76

·7 ·70 10.25

.8 ·71 16.67

.9 ·73 24.24

·95 .74 26.58

1f.l.=

2

In this case, (4.54) gives

Vex (2~') = [,,2(5 - 21» - IjQ. + ~ ~The optimum value of Q is

2Q := (5 - 2p)

We again tabulate the optimum values of Q and the percent gain

in efficiency in Table 4.10.

. Table 4.10.~

Optimum value of Q to be used in the estimator 2T'1 A

when ~ =2 and the percent gain in efficiency over 2T


.6 ·53 5·55

·7 ·55 12·50

.8 .59 21.43

·9 .62 33·33

·95 .65 42.85

66

~Comparing the nature of the gain in efficiency to that of 2T in

Tables 4.2, 4.3, and 4.4, we see that except for p = 0.6, 2~' willlit

produce higher gains in efficiency than 2T (when the corresponding

optimum value of Q is used in each case.)

If the simultaneous optimum values of ~ and Q are required,

~they can be determined by minimizing Var (2T') with respect to Q

and ~ in (4.54) !.~. minimizing

[Q2~2(1_2e) + ~(2Q-l) + (Q-l)21C2(Q,~) = ~(l-~) ] .

Except for some tedious algebra, the procedure is straightforward

and will be omitted.

4.3.3. A General Linear Estimator

Another type of linear estimator which is relatively more general,

can also be constructed to estimate 2T, the total of population char

acteristic of interest for the second occasion. This type of estimator

", 67

was introduced by Hansen et al. [7) in their book and will be referred

to as a general linear estimator. It is given by

(4.55 )

where

-!...Iln N. m lYi,,\ J. (4.56 )T = L: L:1,1 Iln i=l Pi j=l m

is the unbiased estimator for IT based on the first-stage units"A A

appearing from order 1 to Iln, 1,2T, 2,lT and 2,2T are as defined in

(4.14), (4.13) and (4.50) respectively, and a, b, c, d are appropriate

constants to be determined so that 2T* is an unbiased estimator with

the least possible variance. The following diagram is helpful in

.understanding the structure of this estimator:

A A

1, IT 1,2T

,. - ... ~

1st occasion I: jJ.n (l-Il )nt

2nd occasion (l-Il )n Iln

..'" ~

--- -A A

2,lT

2,2T

4.3.3.1. Determination of Constants. As will be evident from the

structure of the above diagram, this estimator 2T* utilizes in a

different way the past and present information available from the

proposed s~ling design.

68

We have

"'* [ 1 f..Ln Ni m lYi oJT =a - L: - L: £.bl2 f..Ln i=l Pi j=l m

and, as noted earlier, the index i in the sunnnation signs refers to

the order of occurrence of each first-stage unit in the sample.

"'*First, we require that 2T shall be unbiased. For this we must

have

i.e.

so that we must have

(a + b) = 0

and

(c + d) = 1

giving

b = -a and d = (1 - c).

Under this restriction, we can now write (4.55) as

(4.58)

(4.60)

69

Next we search for the best combination of values of a and c- -"*in (4.58) so that the estimator 2T will have the least possible

variance. Bearing this in mind, we proceed to determine the best

values of a and e.

From (4.58) we have

Var (2T*) = a2

[var (l,lT) + Var (1,2T) - 2 COY (l,lT, 1,2T)]

2 " 2 "+ C Var (2,lT) + (l-c) Var (2,2T)

+ 2 ac [COY (l,lT, 2,lT) - COY (1,2T, 2,lTj

+ 2a(1-c) [COY (1,1T, 2,2T) - COY (1,2T, 2'2T~

+ 2c (l-c) COY (2,lT, 2,2T) •

In (4.59) it can be shown that

" AThe following five covariances in (4.59), viz. COy (l,lT, 1,2T),

COy (l,lT, 2,lT), COy (l,lT, 2,2T), COy (1,2T, 2,2T) andA ...

Cov (2,lT, 2,2T) can be shown to be zero.

Using some of the previous results from (4.33) we find

Var (2T*) = a2 [~~ (19 + (l-~)n (lVy~ + c

2 (l-~)n (2Vy)

+ (1_c)2 ~ (2vy) - 2ac (l:~)n (1.2Vy)

70

By collecting terms and si~lifying, the expression is

2 V [2 ]v (V)Var ( T*) = . a (~) + (c-l) + 2c~-~ ~ _ 2ac. 1.2 y •2 ~(l-~) n ~(l-~) n (l-~) n.

(4.61)

Setting the f'irst partial derivatives of' Var (2T*) with respect

to a and c to zero, we f'ind after si~lif'ication

~ (IVy) - c (1.2VY) = 0

(C-l)~+/-l) IVy - a (1.2Vy) = 0 •

Solving (4.62) and (4.63) f'or a and c, we obtain:

(4.62 )

where

a =

c =

2~/-lp - /-l P ~2 2 V(l-/-l p) 1 y

(l-/-l)2 2)(l-~ p

(4.64)

as previously def'ined.

To justify that the above values of' a and c minimize Var (/i*),we consider

02 t ~) -2 (1.2Vy)

B = aadc Var (2T*) = (l-/-l ) n ,

02 l ~ J 2 V.- A = oa2 Var (2T ) =

/-l(l-~ )(U)

n,

02 lVar (2T*)}2 V

C = = (~)oc

2 /-l(l-~ ) n

~ 71

Clearly A > 0, C > 0 for all IJ. > 0 and < L

We see that

4 [( V ~2_ (1V;)(2VYJ= 2 2 1.2 Y. 1J.2(l-IJ.) n

= 4[ ~p2 - , } (lVY)(2Vy)J

(l-IJ. )2n2

Since B2

- AC < 0 will imply that a and c given by (4.64)

and (4.65) minimize Var (2T*), we see that the above condition is

satisfied when

2 1 < 0P - 1J.2

or when

or

p < 1IJ.

..-Since 0 < IJ. < 1, it is evident that the above condition is

always satisfied.

72

Having obtained the optimum values of the constants a andc,

we can now write the exact form of what may be called a best unbiased

estimator T* as2

[ A AJ (l-Il)1,lT - 1,2T + 2 2

(l-Il P )

In terms of the observations, the form of this estimator is

(l-Il ) [1 nNi m 2:iJJ+ r:2 2 (l-Il)n i';n+l Pi(l-Il P ) j=l

2 [1 ni;Ln Ni m 2:q ]+ Il(l-p.p ) - r: r:2 2 Iln i=n+l Pi(l-Il p ) j=l

(4.66 )

73

4.3.3.2. A Comment on.the Form of the Estimator 2T*. The

estimator /i* requires knowledge of the values of 2Vy and lVy

and p • This might not be so handy in practice especially when the

values of 2VY and 1VY are not known and must be estimated from the

sample itself. (Usually in large scale sample surveys, the computation

of the sampling variances, if ever they are made, are made long after

the computation of the estimates of the population characteristics of

interest.) However, under the assumption that 2V = lV =V , they y y

estimator will involve only the value of p whose value may be

determined by past experiences or by judicious guessing.

4.3.3.3. Efficiency of the Estimator 2T*. Using the weights a

and c given by (4.64) and (4.65) in (4.61), we have

Var

V1.2 Y

After some simplification, the expression for the variance of

T* is2

Or in a more suggestive form:

A

We will now compare the efficiency of 2T* with 2T for some

values of 1..1. which might be adopted in practice.

Case I:

11..I.=lj:

In this case, Var (2T*) given by (4.61) is

[2 ] V'

Var ( T*) = .1 M 3p ~2 16 ,2 n.p -

= Var ( T) M .2L. (2Vy1 .

2 16Mp2 t n j

A* A

The percent gain in efficiency of 2T over 2T is

2G (p)= (3p ) x 100

6 16;'4p2

and is tabulated in Table 4.11.

A* A 1Table 4.11. Percent gain in efficiency of 2T over 2T for 1..1.. = lj:

p %Gain in efficiency

.6 1.42

·1 10·55

.8 14.28

·9 19.04

·95 21.81

14

75

Case II:

1IJ.=-

3

In this case,

Var ( T*) = [1 _ 2P2J. 2V

y2 9 2 n-P

.....* .....and the percent gain in efficiency of 2T over 2T is

2G (p) = (~) x 100 •7 9-3p2

These values are tabulated in Table 4.12 below •

Table 4.12.

Case III:

.....* ..... 1Percent gain in efficiency of 2T over 2T for IJ. = 3


.6 9·09

·7 13·01

.8 18.08

·9 24.66

·95 28.62

1IJ.=

2

In this case,

A.. [ 2l 2V.. yVar (2T'ft") = 1 - 4:p2j n

76A* A

The percent gain in efficiency of 2T over 2T for this case

is

We again tabulate these values in Table 4.13.

Table 4.13.A* -, A 1

Percent gain in efficiency of 2T over 2T for ~ =2


.6 10·98

·7 16.22

.8 23·53

·9 34.03

·95 41.11

,

A*4.3.3.5. Optimum Value of ~ when the Estimator 2T Is ~sed. It

is of some interest to determine the optimum value of ~ !.~., the

proportion of the first-stage units to be replaced after the first

occasion, and "to see how the gain in efficiency changes when the optimum

~ is used against natural values such as ~, ~ , ~. In (4.67),

setting the first partial derivative of Var (2T*), with respect to ~,

equal to zero we have

[(1_~2p2)(_p2) _ (1_~p2)( _2~p2 )] ~ = 0,(l_~2p2)2 n

or

2 2p ~ - 2~ + 1 = 0

j.l =

yielding the solution

1 !.V1 _ p2

2p

Since the acceptable value of j.lmust lie between 0 and 1, the

appropriate root which gives the optimum value of j.l is

77

= 1 - -V 1 - p2j.l 2

p(4.68)

We compare the gain in efficiency when the optimum value of j.l

is used to the gain in efficiency when ~ = ~;! and ~.

Substituting the optimum value j.l = 1 - 2 1-02 in (4.67) andp

after some simplification, we obtain:

Var ( T*) = [. p2 j 2Vy

2 opt. y 2 n2 - 2 1-p

11/

The percent gain in efficiency of 2T* over 2T when the optimum

j.l is used is

11/It is interesting to note that. there is

Var ( T*).. = ! t. p2 J 2V

y2 opt. 2 .... r--'2 n

- V 1-p'"

where j.l t = the optimum value given by (4.68).op •

a relation

V= -l (~)

2j.lopt. n

,)

..

78

The optimum values of ~ as a function of p and the gain in

efficiency are tabulated in Table 4.14.

Table 4.14. Optimum value of ~ when the estimator 2T* is"used and the percent gain in efficiency over 2T

p Optimum ~ %Gain in efficiency

.6 ·55 11.11

·7 .58 16.73

.8 .62 25.00

·9 .69 39·50

·95 ·77 53.33

Comparing the gain in efficiency 'using the optimum value of ~with

111the gain in efficiency using natural values of ~ = 4' ' 3' ' 2' (Tables

4.11, 4.12, 4.13), we see that the gain does not increase much from

1 1using ~ = 2' .,.. and in view of·· simplicity the sampler may use ~ = 2'

"*without sacrificing much efficiency of the estimator 2T •

Moreover, comparing the extent of the gain in efficiency using

"* ~ ~tthe estimator 2T with that of 2T and 2T for the three cases,

111 tnamely ~ = 4' ' 3' ' 2' ' and when the optimum Q s are used in both

estimators 2~ and 2T' (Tables 4.2, 4.3, 4.4, 4.8, 4.9, 4.10, 4.11,

) A*4.12, 4.13' , we see that the use of the estimator 2T will produce

the largest gain in efficiency.

"*It should also be noted that in comparing the efficiency of 2TA

with that of the simple estimator 2T, no assumption regarding the

79

stability of th~ variance from occasion to occasion is made as in the

former set of comparisons.

4.3.4. A Ratio-Type Composite Estimator

Another type of estimator that might be used to estimate 2T is

given by

where

.A

2~* = 'l[(2,1~){lT)] + (l-'l) [2~1,2T

A A. A A

2,lT, 1,2T, IT, 2T and Q are as defined previously.

A*4.3.4.1. Expected Value of 2T. From (4.69), we have

(4.70)

A

We have shown that E(2T) = 2T. To obtain an expression for

E[:2'1~)(lT)1, we refer to an unpublished lemma given by Koop.

1,2T JLemma I: If X, Y, Z are three random variables and Z:f 0 , then

the following relation holds:

E[X£] = ~[1 + COvx(~,y) COVx(~·Z) -

_ COv)~,:,Z)] + ir- (z : z)21XYZ ~ Z J

where X =E(X), Y = E(Y), Z=E(Z);

and Cov (X,Y,Z) = E[ (X - X)(Y - Y)(Z - Z)]

Cov (Y,Z)

YZ

(4.71 )

80

The proof of this lemma follows from the identity

~ == ~ [1 + 6X + b.Y +~Y - t:fZ, - t:.Xt::Z - b.YhZ - A'X't:.YhZ]z

+ [1m (t:fZ, )2] (4.72)zwhere

!:iZ = (z: Z)Z

We now apply the above lemma to find E[(2'1~) (IT) which is

1,2T Jone component of E(2~)' Recall that E(2,lT) =2T, E(1,2T) = IT

A A

= E(lT) and that 1,2T rO. Hence by the lemma, we have

[

A j [ A ATAT Cov ( T, T)E (2,1) (T) = (~)( T) :1 + 2,1 1

T 1 IT 1 2T1T1,2

81

Or equivalently,

From (4.70) and (4.73), we obtain

.2T A A Cov (211~::' 1,2T, IT)

- 1..fCov (IT, 1,2T) -

1..f

A

IT)2]] •+...1.. E[ :2!1~)(lT) k2T-l..f 1,2T

(4.74 )

~

Hence, 2T"', unlike the other estimators considered previously,

is not an unbiased estimator of 2T. The amount of bias is Q times

the function involving the four covariances, the triple covariance and

the remaining involved product-moment given in the square bracket.

However, upon examining the nature of the covariances (see (4.33)) and

the product-moments involved which are of different signs and are

scaled by the population totals and the square of the totals

respectively, we will see that the amount of bias is likely to be small,

especially when multiplied by Q which is less than one.

l'.:::4.3.4.2. Variance of 2T*. From (4.69), we have

A 2[Var (2T*) = Q :Var

A

To obtain the _essions for Var t(2.1;)(lT)} and.

A 1,2

Cov {(2'1.~)(lT), 2T 1, we will first establish two lemmas which W1ll

1,2T )

be applicable to our problem.

Lemma II: If X, ~ Z are three random variables which are not neces

sarily statistically independent, and Z r 0 , then

._w

Var -- 2 t(XY)'= [XY] Var (X) + Var(Y) + Var (Z) +2·Cov (X,Y)Z Z . X2 ;p. ~ Xy

~-- 2[_ 2 CO~£X,Z) _ 2 CO~ &Y,Z) + [~] Var (8XAY)

XZ YZ Z

+ Var (1SI.l:Il) + Var (6.YM.) :.. Var (8XAYM.)

+ 2 COY {b.X, 6.X6.YJ - 2 COY {b.X, 6.Xb.Z}

- 2 COY tb.X, 6.YM.J - 2 COY (b.X, 8XAY6.ZJ+ 2 COY tAY, 8XAYJ - 2 COy t6.Y, 1SI.l:Il) - 2 COY { bY., 6.YM. }

- 2 COY {AY, ~J - 2 COv( &, 6X6.YJ+2 COY{&,~) +2 COV{&, 6.Y6Z} +2 COV{ &, 8XAY6.ZJ

(continued on next page)

i 83

- 2 COY { AXAY, ~} - 2 COY {AXAY, f:::,.":f.8l, 1

- 2 COY [AXAY, AXA":f.8l,) + 2 COY \ l:!:I.b2., tJf:::,.":f.8l,J

+ 2 COy t=, =)] - Var ~~ (llZ)'

+ 2 COy tr, r (t:::..Z)2]

where

=t:::..X = (X:X), f:::,.Y = (Y:Y), 6Z = (Z:Z) •

X Y Z

Proof: The identity (4.72) shows that

n ~[ ]Z == -; 1 + t:::..X + f:::,.Y + t:::£L:::i - 6Z -~ - &M., - AXA":f.8l,Z

+ [~ (6Z)2)

(4.76)

Transposing the last term. on the right hand side to the left hand

side, we have

n n( )2 XY[- - - til =- 1 + t:::..X + t::J - til +~ - !SXAZ - f:::,.":f.8l,Z Z =Z

84

• • Vax [~] + Vax [~ (AZ)2] - 2 Cov [~ , ~ (AZ)2]

= [~t [vax (LIX) + Vax (£:;Y) + V';" (lIZ) + Vax (IIUIY) + Vax (LIXAZ)

+ Vax (6.Yb.Z) + Vax (1Sf.6.Yb.Z) + 2 Cov t6.X, 6.Y) - 2 Cov \ 6.X, t1l.J

+ 2 Cov t6.X, ISf.6.Y) - 2 Cov {6.X, 6.XbZ) - 2 Cov t6.X, 6.YDZ)

- 2 Cov {6.X, 6.XAYAZ) - 2 Cov t6.Y, AZ} + 2 Cov ( 6.Y, ISf.6.YJ

- 2 Cov t6.Y, 6.X!:fl..) - 2 Cov {6.Y, 6.Yl:1l} - 2 Cov { boY, ISf.6.YDZ)

- 2 Cov tAZ, ISf.6.Y) + 2 Cov {AZ, 6.XbZJ+ 2 Cov { AZ, 6.Yb.ZJ

- 2 Cov t1:::.X6.Y, 6.X!:::..Yb.Z) + 2 Cov t6.XbZ, 6.YDZ}

+ 2 Cov t6.Xtfl, ~)' + 2 Cov t1Sf.t:jZ, D.XAYDZJ].Now Var (6.X) = Var (X:X) = ~i Var (X)

X XSimilarly,

Var (6.Y) 1= =2 Vax (Y) ,

Y

Var (t1l.) = ~ Var (Z) •

Z

Cov t6.X, 6.Y) = Cov {(X:X), (y:y)} = =: Cov {X,Y),X Y- XY

85

and

... =: Cov tX,Z ~ ,xz

=: COv { Y,Z) .yz

. . from (4.77), we will have

+ [~l2 [var (=) + Var (=) + Var (l>YllZ)

+ Var (A':l.AY6Z) + 2 Cov {8X, A':l.AY) .. 2 Cov {8X, b:.XAZ }

.. 2 Cov (8X, l:::.Y6ZJ .. 2 Cov {8X, A':l.AY6Z}

+ 2 Cov tl:::.Y, A':l.AY) .. 2 Cov {l:::.Y, b:.XAZ )

+ 2 Cov { 6Z, A':l.AY6Z} .. 2 Cov lA':l.AY, t:::.Xt:::.ZJ

.. 2 Cov tA':l.AY, l:::.Y6Z} .. 2 Cov tA':l.AY; t:::.Xt:::.Y6ZJ+2 Cov(~,ISl6Z)

+ 2 cov {b:.XAZ, t:::.XAY6Z) + 2 Cov tb.Y6Z, A':l.AY6ZJ]

- Var [~ (t,Z)2] + 2 Cov [~ , l>f (t,Z)2J •

86

Lemma III: If X, Y, Z and W are random variables which are not

necessarily statistically independent and Z r 0 , then

(4.78}

COY [~ ,wl = X! [COy)X,W) + COY)Y,W) _ cov)z,w)lZ . X Y Z J

+ ~ fCoV (=, II) - Cov (=, II) - COv (EfAZ, II)

- Cov (=t1Z, II)] + Cov [~ (l>Z)2, II]

Proof: Using identity (4.72), we have

Cov [(~), III = COv[(~ (1 + 1St. + l!>f += -l>Z -=- IYUSYN,) + ~ (l>Zf} , II] • (4.79)

Hence, the R.H.S. is

[~l (COy (~,W) + COy (~Y,W) + COy (~Y,W) - Cov (~,W)

- COV (=,11) - Cov (6YllZ,II) - COV (=t1Z,tn]

+ COy t~ (~)2, W1 0

Since COy (~,w) = COy ((X~X) , W}

1 {- = J= X l E[(x-x)wl - E(X-X) E(W)

1:; E (xw) - X E(W)X

= : E (xw) - E(X)E(W)X

= ~ COy (x,w) 0

X

f 87

Similarly,

.... COY (AY,W) 1= -;- Cav (Y, W) ,Y

1COY (~,W) =-=- COY (Z,W) 0

Z

Hence, from (4.79)

COY [(~) , W] = X! [cov_(X,Wl + cov)Y,W) .. cov).z,W)]z X Y Z

+ X! [COY (AXAY,W) .. COY (AXAZ,W) .. COY (AYM.,W)Z .

.. COY (Az/W/\z,W)] +COV t~ (~)2., W) '0

In applying the two lemmas to our problem where the various

estimates under study play the role of random variables, we will use

only the leading terms, !o!o, the terms of second order or less in

(4076) and (4078) by assuming that the magnitudes of the remainder

terms in the two formulae are relatively small as compared to the

leading terms 0

Thus, the approximate expressions for

"'-

and Cov ((2., 1~ )(1T), 2.TJ1,2

T.

are:

88

and

+ Var (1,2T) + 2 Cov (2.z'lT, IT)

I T2 2TIT

2 OOv (:~ l}l] (4.'80 )

(4.81)

A

Hence, from (4.75), the approximate expression for Var (21!*) is

A 1Using previous results such as Var (2,lT) = (l-lJ.)n (2Vy)'

" " 1Cov (2,lT, 1,2T) = (l-lJ.)n (1.2Vy) etc., we can write the approximate

89A

expression for Var (2T*) as

~ 2[ 1 2T 2 1V 2T 2 1Var (2T*) ~ Q(l-~)n (2Vy) + (~) ~ + (~) (l-~)n (lVy)

1 1

22T 12T 1 2T 1 ]

+ 2 (IT) n (1.2VY) - 2 (IT) (l-~)n (1.2VY) - 2 (iT) n (lVY~

+ (1_Q)2 ! ( V ) + 2Q(1-Q) r! ( V ) + (2T

) (l-~)n (' V )n 2 y In 2 y 1T n2 1.2 y

Or, by collecting some terms, we have

where

R =

Remarks:

[2 2 2]+ 2R .Q~ - Q~ - Q ~

. (l-~)

V1.2 yn (4.82)

(i) Comparing (4.8,2) with (4.35), we see that the approximate~

formula for Var (2T*), is very similar to the exact formula forA

Var (2T) !.~., the variance of the linear composite estimator, except2T A A A .

for the ratio R = T· When E(2· IT) = E(lT) = E(l 2T) !.~.,1 ' ,

when 2T = IT the two formulae are identical.

(ii) The approximate formulae (4.80) and (4.81) ca,n also be

derived by the use of the Taylor approximation technique, ! ..!., by

.. 90

expanding

'"= (2,1T) ( T)

'" I1,2

T

into a power series about the point (2T, IT, IT). Neglecting the

terms of powers higher than one, we get

A

(2,1~)(lT) ~1,2T

(.2T

T)( IT ) + (. T- T} 0

1 2,1 2 0 T2,1

A

{ (2,1~)(lT)J1,2T

+ {T- T} l1 1 ~ A

°IT

which gives

A

IT':"IT

or

A

2 IT A {A} f A, } 2T

{ A J(~)(IT) s. 2T + 2,lT-2T + IT-IT -T - 1 2T-IT1 '1,2T

(4.83)

91

andfrom which the approximate formulae for Var {(2'1~)(lT)}1,2T

Cov { (2'1~)(lT), 2TJ can be arrived. However, the latter technique

1,2T

must rely on the assumption of differentiability near (2T'lT, IT)

and the existence of partial derivatives Which cannot be guaranteed

simply because the estimates involved are not continuous.

(iii) From (4.83), we also note that when 2T = 1T , we will have

'" '" '"IT + 2,lT - 1,2T

which in turn, implies that in such case

!.~., the ratio-type composite estimator will give approximately the

same estimate as does the linear composite estimator.

~*4.3.4.3. Efficiency of 2T. From (4.82), we see that the

A 2Ta.pproximate variance of 2T* involves the ratio R =T which is

1

unknown. However, in practice, we may use the sample ratio

as an estimate of R.

By rewriting (4.8,2), we have

•

Under the assumption that V = V = V and writing p V forly 2y Y Y

l.2Vy , we have

Var (2~*) ~ Var (2T) + ri (1~~) [1 + ff] ~

[2 2 2] V+ .. QJl - QJl - Q Jl .1l.

2 P R . (l-Jl) n •

From (4.84), we see that when p = 0 , we will have

(4.84)

which implied that in this case (!..!., no correlation over time), the

ratio-type composite estimator to the order of approximation involved,"'-

is less efficient than the simple estimator. 2T. In practice, we

would expect the correlation p to be sufficiently high so that the

use of the estimator 2~* will result in a gain in efficiency over theA

estimator 2T.A

Since the expression for Var (2T*) given by (4.84) does involve

2Tthe value of the ratio R =~ , we cannot make any specific comparison

~ 1about the efficiency of 2T*. However, we will examine the nature of

the interval value of R, which is usually unknown, for some specific~

cases of interest in which the use of the estimator 2T* results in aA

gain in efficiency over the simple estimator 2T •

We consider (4.8'4). To the order of approximation involved,

* "'-Var (2T*) will be less than Var (2T) if

Q2fJ. f- l + R21 ~+ 2 p R [QJl2

- QJl- Q2Jl21 ~ < 0(l-Jl) t J n . (1-fJ.) J n .•

93

For 0 < Q < 1, 0 < IJ. < 1, the above condition is equivalent to:

Specific cases for

* AThe estimator 2T* will be more efficient than 2T if

(ii)

A~ ~The estimator 2T will be more efficient than 2'1' if

~ [5p - -J25p2 - 9 ] < R <.~ [5p +-J 25p2 - 9 ]

1 1( iii) For Q =2" ' IJ. =2" •

~ A

The estimator 2T* will be more effiOient than 2T it

4.4. Estimation of Change in Total between the First andSecond Occasion

To estimate the change in the total of the population characteristic

=

94

several estimators can be constructed. To this end, we will consider

the estimators which bear close resemblance to the estimators used for

estimating the total. Also, attempts will be made wherever possible

to compare the relative efficiency of these estimators with respect to

the simple estimator

(4.86)

4.4.1. An Estimator Based on the Linear Composite Estimator

4.4.1.1. Estimator. One possible estimator that can be used to

'"estimate Dl is Dl whose structure is:

'"~ = T-1T1 2

'"'"where 2T is the linear composite estimator as defined in (4.12).~ A

Since we have seen that E(2T) = 2T and E(lT) = IT , this

implies that

= T - T.2 1

(4.88)

95

Using the results obtained previously for the relevant

variances and covariances in (4089), we Will have

Under the assumption that

and by writing p VY for 102VY , the expression for the variance of~Dl becomes

;... 2[ Vl 2[ ]VVar (Bl ) = Q 2(1-p) (l-~)rij + 2(1-Q) 1 - (l-~)p ++ 2Q(1-Q) [2(1-P) V~] ,

which can be rewritten in a more suggestive form as

V(.L..)

n

(4.91)

..

96~4.4.1.3. Efficiency of Dl • To see how the use of the estimator

AA ADl will result in a gain in efficiency over the simple estimator D

l,

we recall that

with

=! ( V ) +! ( V ) _ 2(1-~)n2y nly n (4.92)

we will haveUnder the assumption that IVy = 2VY =Vy

Vax (~) =2 [1 - (l-~)pJ V~ •

Hence, from (4.91) and (4.93), we have the relation

~We see again that when p =0 , the estimator Dl is less

A

efficient than the simple estimator Dl , for in such a case

V

(l~~) (+)

,.,.

since the second term is always a positive quantity.

However, when the correlation p is high, we would expect that the~

use of the estimator Dl will give a more precise result. For this,~ A

we will now examine the gain in efficiency of Dl over Dl for some

values of ~ and Q Which might be adopted in practice.

97

Case I:

Here we have

Var (~ ) =Var (D ) + (4 - 13p)1 1 24

v...L

n

~

The percent gain in efficiency of D1 over

( ) _ ( 13p - 4)gl P - 52 _ 49p x 100.

*The percent gain in efficiency of D1

shown in Table 4.15.

....over D1 for this case is

...

-

A ....Table 4.15. Percent gain in efficiency of D1 over D1 when

1 11-l=4' Q=2'

p . %Gain in efficiency

.6 12.39

·7 28.81

.8 50.00

.9 97.47

·95 153·21

Case II:

In this case, we will have

.. v...L

n

..

A ~

The percent gain in efficiency of D1 over D1 is

( ) _ ( 9p-3 )B2 P - 27-25p x 100 •

Table 4.16 shows the nature of the gain for some values of P

which may arise in practice.

98

Table 4.16.

Case III:

~ "-Percent gain in efficiency of D1 over D1 for

1 11-1=:3' Q=2'

P %Gain in efficiency

.6 20.00

·7 34.73

.8 60.00

·9 113·33

·95 170·77

\

1 11-1=-, Q=-2 2

In this case

90 A

The percent gain in efficiency of D1 over D1 is

( ) _ (3) -2 )g3 P - 10-9p x 100.

Table 4.17 shows the nature of the gain in efficiency for this

case.

Table 4.17.~ A

Percent gain in efficiency of D1 over D1 for1 1

1-1=2"' Q=2"

P %Gain in efficiency

.6 21.74

·7 40.34

.8 71.43

·9 131.58

·95 189.65

99

4.4.1.4. A Comment about the Gain in Efficiency. Comparing the~ A ~

nature of the gain in efficiency using D1 over D1 to that of 2TA

over 2T for the three cases considered, we see that sUbstantial gain

will be realized in estimating the change. In practice, we may have

to estimate both the change and the current tot.a1. This should present~

no problem for once 2T is computed,~ .

D1 is automatically obtainedA ~

by simply subtracting 1T from 2T. To see how the gain in effi-

ciency changes when the optimum value of Q is used in (4.88), we

will now consider the gain in efficiency for the three cases

considered previously.

Case I:

1 ~When 1-1 = 4' Var (D1 ) as given by (4.91) is

~ [ 2fL 2 2 (Q2 4tlQ , 1 VvVar (D1 ) = 2 (1 - T) + 3' Q - - 6 ~ Pj ~

~MinimiZing Var (D1 ) with respect to Q, results in the optimum value

Q =..2L-(4-p)

..

100

When this optimum Q is used we will have

A

The percent gain in efficiency over D1 is

3 2'::7..,....lL.... x 100.10 (l-p)

The percent gain in efficiency is tabulated in Table 4.18.

Table 4.18.

Case II:

A

'"Optimum value of Q to be used in the estimator D1

andA 1

percent gain in efficiency over D1 for ~ =4


.6 ·53 16.87

.7 .64 30.62

.8 .75 60.00

.9 .87 151.87

·95 ·93 338.43

When 1~ = 3 ' (4.91) leads to

~ [ 2 2 Q2 + 4Q JVVar (Dl ) =2(1 - 3 p) + Q - 3 p ~.

~Minimizing Var (D

1) with respect to Q, results in the optimum

value of Q = (3:P) ·The percent gain in efficiency over

101

The optimum values of Q and the percent gain in efficiency over....D1 for this case, are tabulated in Table 4.19.

....Table 4.19. Optimum value of Q to be used in the estimator 1)1 and

percent gain in efficiency over 1)1 for ~ =~


.6 ·50 20.00

·7 .61 36.29

.8 .73 71.11

·9 .85 180.00

·95 ·93 401.01

Case III:

1When ~ =2:' (4·91) gives

VVar (D ) = [2(1 - 2.) + 2Q2 .. (Q2 + 2Q)p] J:.

1 2 n

and the optimum value of Q is ~ •

'"The percent gain in efficiency over D1 is

x 100.

The optimum values of Q and the percent gain in efficiency over

'"D1 for this case, are tabulated in Table 4.20.

102AA

Table 4.20. Optimum value of Q to be used in the estimator Dl andA 1

percent gain in efficiency over D1 for ~ =2


.6 .43 22·50

·7 .54 40.83

.8 .67 80.00

.9 .82 202·50

·95 ·90 451.25

4.4.1.:2. A Remark about the Optimum Value of Q. It should be

:::noted that although the estimator Dl is based on the linear composite

~ ~ A

Dl = 2T- 1T , the optimum values to be used in

1 ~and 2' the optimum value of Q for Dl

AA

optimum value for 2T •

1= 2 ' the optimum value of Q to be used in

A

the optimum value to be used in ~l is

~

estimator 2T, !.~.,

~ ~

2T and Dl are~ the

1 1namely, when ~ = '4 ' 3'turns out to be twice the

For example, when ~

~ whereas2(2-p) ,

same. In the three cases which we considered

,

In practice, the sampler must have the primary objective in mind

whether he wants to get an estimate of the total or the change in total

and choose the best weight which will result in the best gain. How-

ever, when the sampler wants to get good estimates of both the current

total and the change in total between the two occasions, the appro-

priate weight Q for estimating total may be used in both cases without

sacrificing much efficiency.

..103

4.4.2. An Estimator Based on the Modified Linear COmposite Estimator

Another possible estimator which can be used in estimating Dl

is'"

B~ = 2~' - IT (4.94 )

~,

where 2T is the modified linear composite estimator as defined in

(4.49) .

We have seen that

evident that

'"""!.~., D is also an unbiased estimator of Dl •

..By writing (4.94) as

or

we find

,)

(4.96)

104

Remembering that

A A

COV (2,2T, IT) = o ,

A A

COV (2,lT, 2,2T) = o ,

A A

COV (1,2T, 2,2T) = o ,

and using the expressions obtained earlier for the relevant variances

and covariances, we will have

+ [l_Q]2[..1..( V ) +! ( V )1j..I.n 2 y n 1 y J

+ ~ (1Vy )] •

+ 2Q( l-Q) [- 1 ( V )n 2 y

Under the assumption that 2Vy = lVy = Vy and L2Vy = pVy' the

expression now becomes

'" [ 2 2 2 2] VVar (n') = (l-Q) -l-Q j..I. + 2Qj..I. - j..I. ..JL +1 j..I.(l-j..I.) n

..

~I4.4.2.2. Efficiency of Dl • We will now consider the efficiency

of the estimator ~:i. relative to the siIll.Ple estimator £1 for some

specific cases which might be adopted in practice.

Case I:

105

In this case

Var (E~) = [~ - t p] v~

= Vax (D ) + [....2 + ! pJ :.z..1 J2 3 n'

where

var(l\)= [2 -i p] V~ , as given by (4.93).

A A ~,

Hence Vax (j)i) > Var (Dl ) for all °< P < 1 !o!!.. Dl is~

efficient than the simple estimator j)lo

Case II:

1IJ. =3' '

In this case

Var (~~) = [t -t p] +

> Var (Dl ), for °< P < 1 °

~

Hence, we see again that in this case, Di is also~ effi-

'"cient than the simple estimator Dl °

Case III:

In this case,

Var (~i) = Vax (Dl ) +~ V~

106

~IA

The percent gain in efficiency of Dl over Dl is given 'by

g7(P) := (2p;S1) x 100 .9- P

A AAI

The percent gain in efficiency of Dl over Dl for this case

is tabulated in Table 4.21.

Table 4.21. Percent gain in efficiency of ~~1 1

~=2"' Q=2"

A

oyer Dl for

..

P %Ga~n in efficiency

.6 3·70

.7 8.33

.8 14.30

·9 22.22

.95 27·27

We see from the above table that the amount of gain in efficiency

is not substantial.~t

To improve the efficiency of Dl for the three

cases considered above, the optimum weight Q for each case may be

used. Since the procedure is analogous to that used in the estimator~

Dl , we willmt present the details of the derivation, but only the results

will be given. These are found in Table 4.22.

A comment:At

Comparing the nature of the gain in efficiency of Dl to

.'

----~

~that of Dl when the optimum values of Q are used in the three

111cases namely ~ = 4 ' :3 ' 2' ' we see that the gains from using the

~t Aestimator Dl are slightly~ than the gain from using Dl • This

is due to the structures of the two estimators themselves. (See (4.88))

107~

Table 4.22. Optimum values of Q to be used in the estimator D''"

1and percent gain in efficiency over D1

Case I: 1 Optimum Q = (3p + 12)1l=1j: 17 - 2p

P Optimum Q %Gain in efficiency

.6 .87 11.98

·7 .90 26.34

.8 ·93 56.34

·9 .97 148.86

·95 .98 336.15

Case II: 1 Optimum Q = t; ~ pjIl=-3

.6 .81 13.79

·7 .86 30.67

.. .8 ·90 66.10

·9 ·95 173·10

·95 ·97 393.33

Case III: 1 Optimum . - f2 + p)Il=-2 Q -5 - 2p)

.6 .68 14.65•

·7 .75 33·33

.8 .82 72.88

·9 ·90 195·80

·95 .95 425.14

108

A "and (4.95». The difference 2,2T - IT in (4.95) provides l~ss

"information about 2T - IT than the difference 2T - IT in (4.88).

4.4.3. A General Linear Estimator

Another type of estimator Which can be used to estimate

A ,,~ A

where 1,lT, 1,2T, 2,lT and 2,2T are as defined previously, and,A

a', b', e', d' are constants to be determined so that Dr is an

unbiased estimator of Dl having least possible variance.

Determination of Constant a', b', c'; d'.

to be unbiased we must have E(Dr) = 2T - IT •

E(Dr) = (a' + b' )IT + (c' + d' )2T •

Imposing the condition of unbiasedness we have

A

For nr

(a' + b' ) = -1

(c'(4.100)

+ d') = 1

giving

b' = -(l+a' ) ,d' = (l-c') ,

so that

Dr = a' \l,lT} - (l+a') {1,2T} + c' t2,lT} + (l-c') [2,2T}.

(4.101)A*We now choose a' and c' in (4.1-01) so that Var (Dl ) is a minimum.

, 109

First we have

(4.102 )

Under the proposed sampling design, the only non-zero covarianceA A

is Cov (1,2T, 2,lT) •

Using previous results for the relevant variances and covariances

in (4.102), we will have

Var (n*) = (a l )2 (lVY

) + (1+a,)2 (A-) + (c,)2 (A)1 Iln· (l-ll)n (l-ll)n

v V+ (1_c,)2 (~) - 2(1+a') c' (~) •

Iln (l-ll)n

Taking partial derivatives of Var (D"~) with respect to a' and c'

and equating the results to zero, we have

(4.104)~ai Var (D~)

~Ci Var (Dt)

= [2a' + «1)' )1 V 2c' ( v) - 0~ 1-1l n] 1 y - (l-ll)n 1.2 y -

[2c' 2(1-c' )1 «1)') ( )

= (l-ll)n - Iln j 2Vy - l-~ n 1.2Vy =0 •

(4.105 )

..111

Using these weights in (4.101), we can now write the estimator as

D* =1

+

+

~l-~) + P~(l-~~fi J[ (1-11 p) Jp~(l-~) -Ji§;. + (l-~)l

(1_112p

2 ) ]

~_~2p2 _ p~(l-~) -J..1Vy

2V

y

(4.110)

"'*4.4.3.2. A Comment about the Form of the Estimator D1 • We see

again that the estimator D~ like the estimator 2T*, requires the

values of 1Vy and 2Vy and p. Under the assum,ption that

1VY = 2VY , the ratios under the radical signs in (4.110) become unity,

so that only the value of p is required in the weight function. It

may also be noted that these variances and p can be estimated from

the samples for two successive occasions as pointed out in Section

4.4.3.3. Variance of D~. Using the value of a', 1 + a', c'

and 1 - c' given in (4.106), (4.108), (4.107), (4.109) in (4.103),

we have

112

-J§. 22 2pl-l(l-l-l) =..Jl. + l-l P - l-l

lVyJ, A lVyVar (D~) = 2 2 l-ln(l-l-l p)

~2

(l-l-l) + Pl-l( l-l-l) =-.ylVlV

+ Y(l-l-l)n2 2

(l-l-l P )

l~(l_~)-J11 + (1-~) J2

V+ 2 Y

(l-l-l)n(1_l-l2p2 )

22 ~2

l-l - l-l P - pl-l(l-l-l) ~2Vy2V

+ y2 2 l-ln(l-l-l P )

2 2(l-l-l P )

After some algebraic manipulations, the exPression is found to be

A [2~V [2~VVar (D*) = l-p l-l !...y +l-p l-l U - 21 122 n 122 n-P l-l -P l-l

ii

......*4.4.;.40 Efficiency of the Estimator Dlo We will compare the~ ......

efficiency of Dl with that of the simple unbiased estimator Dlo

Under the assumption that 2Vy = lVy = Vy , we will have from

(4.111)

113

(4.113 )

We recall from (4.93) that

Hence we have the relation:

Var (Di) := Var (Dl

) _ 2 fp2J.1 - P:J.1

2]. V~ •

l 1 - P J.1

The percent gain in efficiency of Di over Dl as measured by

is

Var "'- "*(Dl ) - Var (Dl )

Var (D~)x 100

(4.114)

"* "'-The nature of the gain in efficiency of Dl over Dl is~ A

identical to that of Dl over D1 for the corresponding cases

111J.1 := 4' ' 3" and 2' when the respective optimum values of Q are used.

1We verify and find that when J.1:= '4

1 _ 3p2g(p, 4') - 16(1-p)

showing the same gains in efficiency as in Table 4.18. The cases when

J.1 := ~ and ~ can also be verified and we will find that

11)g(p, 3") = g5(P), g(p, 2')= g6(p •

"

114

The explanation: for this state of affairs is because when in Dr(as given by (4.110)) we set ,.,V = lV = V , we find the resulting

~ y y y~

form identical to D1 , as obtained when optimum values of Q are

used as weights i~ (4.88), and writing also

4.4.4. An Estimator Based on the Ratio-Type Composite Estimator

Another type of estimator that can be used to estimate Dl =2T - IT

A*is Dl given by

~~ = 2~* - lT

{2.1~)(lT)] + (l-Q) [2T] - " (4.115 )= IT

1,2T

4.4.4.1- A* With the result at (4.74) we findBias of Dlo

so that the amount of bias of

same as the amount of bias of

~*D1

as an estimator of 2T - IT is the

"'*2T as an estimator of 2T °

115

~* ~* ~* A4.4.4.20 Variance of Ill' We have from Dl = 2T - IT ,

A*The approximate expression for Var (2T ) has· been established in

(4.83 ) • The expression for Var (1T) is well known. To find the

A* Aapproximate expression for Cov (2T , 1T), we first recall that

2~ ~ Q r(2.1~) (1;) + (l-Q) [2T] •L1,2T J

Hence

(40116)

Now, to find the approximate expression for

we apply Lemma III. Using only the leading terms,

mate expression to be

A

2 IT A A

Cov [(~)(lT), lTl,1,2T .

we find the approxi-

(4.117)

116

Using previous results for the relevant expressions in (4.117),

we have

l~(4.118)

..

2 1TA A

Substituting the approximate expression for Cov [(~)(lT), 1T]

1,2T.A A

and the expression for Cov (2T, 1T) obtained earlier in (4.116) we

have

A

And hence With (4.83) and recalling that Var (1~)

~*the approximate expression for Var (D1 ) as

[2 2 2] V+ 2R .QIJ. - QIJ. - Q IJ. ~ +!. (V)

(1-1J.) n n 1 y

_ 2Q ( V) _ 2(1-Q) [(1-1J.) ( V)] ..n 1.2 y n 1.2 y

(4.120 )

Under the assumption that 1V = 2V = V , we obtain, after somey y y

simplification:

.....- where

~* V 2 2 VVar (D1 ) = 2[1 - (l-lJ,)p] ~ + Q (1~1J.) [1 + R ] ~

- 2 [f~~~) R + (l+R)~] P ~ (4.121)

117

**4.4.4.3. A Remark about Var (Dl

). Comparing the approximate

~ ~variance of lJl given by (4.121) to the exa.ct variance of Dl given

2Tby (4.91) we see that when R = T' the two expressions are the same.

A 1 VRecalling that Var (Dl ) = 2 [1 - (l-~)p] .:;, we have the relation

Var (~~) ~ Var (D1 ) + Q2 (l~~) [1 + ~] ~

•

~* AIn (4.122), it is evident that when p = 0, Var (Dl ) > Var (Dl ).

When the correlation p is sufficiently high, we would expect that

A*the estimator Dl when used, will result in some gain in efficiency

A **over the simple estimator Dl • However, since Var (D1 ) involves the

2Tratio R = T which is usually unknown, we cannot make any specific

1comparisons regarding the efficiency of this estimator.

We will examine the nature of the interval values of R where

**the use of Dl will result in a gain in efficiency.

:::::* "Now, to the order of approximation involved, Var (Dl ) < Var (Dl )

if

V VQ2 (l~~) [1 +~] 1- (i~) [Q~R + (l+R)(l-~)] P ~ < 0 ,

or if Q[l + R2] - 2[Q~R + (1 + R)(l - ~)] p < 0 •

Case I:

~* AHere the estimator lJl is more efficient than Dl if

~ [7P -1/49p2 + 48p - 16] < R < ~ [7P +-J49p2 + 48p - 16 ].

•

Case II:

118

~* AHere the estimator D1 is more efficient than D1 if

Case III:

Here the estimator ~~ is more efficient than £1 if

•

•

5. MORE THAN 'IWO STAGES OF SAMPLING AND MORE THAN 'lWOSUCCESSIVE OCCASIONS

5.L General Remarks

The estimation theory covered so far has been for two-stage

sampling for two successive occasions. When the number of stages of

sampling and the number of successive occasions are more than two, the

algebra becomes more involved. We will now consider the extension of

the theory to such cases. For this!J we will consider the two exten-

sions of the theory separately, first the theory for more than two

stages of sampling on two successive occasions, and then the theory for

more than two successive occasions where the sampling may be done in

two or more stages. .

5.2. More Than Two Stages of Sampling

When the practical situation demands that the sampling must be in

three or more stages and the sampler still wants to incorporate the

technique of partial replacement of first-stage units to such multi-

stage sampling design for the reasons described in Chapter 1, the basic

scheme as described in Chapter 3 is still applicable. The only addi-

tional thing to be considered is the appropriate method of selecting

units in the succeeding stages. Since the first-stage units are drawn

with unequal probabilities and with replacement and the second-stage

units are drawn with equal probabilities and without replacement, to

accomplish the purpose of spreading the burden of reporting among

respondents, the selection of units in the third and other succeeding

stages may be done as in the second stage, 1.~., selecting the units in

119

120

the third and other succeeding stages with equal probabilities and

\l'

without replacement. As an illustration of the general problem, the

occasion.

extension of the theory to four-stage sampling will be considered.

ththe variate value of the £ fourth-stage unit of

unit of the jth second-stage unit of the i th first-

Let tYijk£ beththe k third-stage

stage unit on the tth

We are interested in estimating

N Ni ~j NijkT = ~ ~ ~ ~ lYo °k O

1 i=l j=l k=l £=1 J.J ~

(i=1,2, ••• N; j=1,2, .•• ,Ni ; k=1,2, ... ,Nij , £=1,2, ... ,Nijk ).

The procedure for selecting units is as follows:

(i) The n + (p-l)~n first-stage units for the preliminary sampleN

are selected with probabilities Pi > 0 (~ Pi=l), and with replacei=l

ment after each draw and the order of appearance of the units noted,

as described in Chapter 3.

(ii) m second-stage units are independently selected with equal

probabilities and without replacement after each draw in each of the

first-stage units selected.

(iii) r third-stage units are selected with equal probabilities

and without replacement after each draw in each of the second-stage

units selected.

(iv) q fourth-stage units are selected with equal probabilities

and without replacement after each draw in each of the third-stage

units selected.

An unbiased estimator for lT based on the first n first-stage

units of the set of n + (p-l)~n units is

121

(5.2 )

where the index i again refers to the order of occurrence of the

first-stage units in the preliminary sample.

'"We show that E(lT) = lT •

By a well-known theorem on conditional expectation

E(lT) = E [ES f ES \ ES (IT)} ;]lt 1,2 1,2,3

where in (5.3) ES ( . ) is the conditional expectation of the1,2,3

function represented by ( • ) given the relevant third-stage, second-

stage and first-stage unit, and so on.

Now

'" 1 n N. m N.. r Ni'k (] Nijk· 1E ( T) = - 2: ~ 2: ...hl. 2: ~.. 2: - y81,2,3 1 n i=l Pi j=l m k=l r q £=1 Nijk 1 ijk£

m Not;2: ....bL

j=l m

where

Further,

1n N, m

=- 2: 2:, 2:n i=l Pi j=l

Ni.J. Ni , 1r. 2: J - lYiJ'km r k=l Nij

where

Again further

122

where

And hence,

,

+s {~ (ES (IT)n]N lYinl: Pi=

1 1,2 1,2,3 n i=l Pi

N N~

Ni Nio k= l: 1Yi = l: l: l: JlYijk.e = lT •

i=l i=l j=l k=l ,£=1

To find the expression for Var (IT) the theorem on conditional

variance is applied. In our case we find

,

where VarS (.) is the conditional variance of the function1,2,3

represented by the dot given the relevant first-stage, second-stage

and third-stage unit, and so on.

We proceed to evaluate each part of the variance.

123

Next

~1 n Ni m ]

= Var - L: - L: Y81 n i=l Pi j=l 1 ij

1 n N. 2= 2 L: (pJ.) var(lYi)

n i=l i

1n N 2 0-2 (N.-m)

= _ L: (-i) 1 i ..,...=.J.~n2 i=l Pi m (Ni -1)

,

where

(lYij= 2

2 ~ - 1Yi) = Ni 1Yij10-i = 1Yi = L:Ni

, Nii=l j=l

Hence

Next, we consider

124

From (5.4), we have

1 n Ni

2 m Ni . 2 _=2 L: (p) L: (..2:.sl) Vars (lY' j )

n i=l i j=l m 1,2 J.

where

Hence

Finally, it can be shown that

125

E [ES {ES ( VarS (1T)}}]"1 l 1,2 1,2,3

= 1 ~ Ni ~i Nij ~ij ~jk l(j~jk (Nijk-q)

n i=l Pi j=l m k=l r q (Nijk-l)

where

= ~ijk£=1

= 2(lYijk£ - lYijk)

Nijk

....Combining these results, we obtain the expression for Var (IT) as

I N lYi 2 lNi liiVar (IT) =- E P. (- - T) +- E -1.

n i=l J. Pi 1 n i=l Pi m

(Ni-m)

(N.-l)J.

1 N N. N.+ - E .2:. EJ.

n i=lPi j=l

Generally for k stages of sampling, the total variance of a linear....

estimator such as 1T will be made up of k parts or components.

To estimate

..

!.~., the total of the population characteristic of interest for the

second occasion when ~n first-stage units have been partially replaced

as described in Chapter 3, four types of estimators namely 2~' 2~t,

....*. A*2T and 2T , can be used, and the theory is the same as in the two-

stage sampling case discussed therein.

N r:f. N.+ .E ..1:. .EJ.

i=l Pi j=l

..

126

As can be seen from the derivation of Var (IT), all other variances

and covariances involved in the four estimators mentioned above, will

be made up of 4 components.

For example,

1 [N 2Y

i lYi= (l-~)n .E Pi (-p- - 2T)(-p- - IT)i=l i i

(2Yij - 2Yi)(lYij - lYi ) 1 (Ni-m)Ni m (Ni-l)

+ ~ Ni ~i ~j ~ij (2Yijk- 2Yij)(lYijk- lYij) 1 (Nij-r)

i=l Pi j=l m k=l Nij r (Nij-l)

+ ~ Ni ~j Nij ~ij ~.llii=l Pi j=l m k=l r

(5.13 )

To estimate the change between the first and second occasion,

!..~. ,

four estimators: can also be used. The theory will

again, be very similar to the theory for the two-stage sampling case.

The changes will be entirely in the structure of the estimators and the

addition of two extra components of variances or covariances as the

case may be.

5.3. More Than Two Occasions

t

•

The estimation theory will be extended to the general case when

multi-stage sampling is carried out on more than two successive occa-

sions and the partial replacement of first-stage units is as described

in Chapter 3. We will first review briefly the scheme of partial

replacement first-stage sampling units and also introduce the notation

to be used.

Suppose that the sampling is done for a successive occasions

where a > 2. On the a th occasion, the n first-stage units to be used

are those units which occur from order (a-l)lln+l to n+(a-l)lln in the

preliminary sample. Of those n first-stage units, there will be

(l-Il)n first-stage units occurring from order (a-l)lln+l to n+(a-2)lln

common to the (a-l)th occasion; the other Iln first-stage units which

occur from order n+(a-2)lln+l to n+(a-l)lln are newly selected to

replace those Iln first-stage units which Occur from order (a-2)lln+l

to (a-l)lln. (See diagram.)

order: (a-2)lln+l (a l)lln+l n+(a-2)fln

~~:onI--+----!----III---·_ath occasion ..

The sub-units in the succeeding stages are selected in the manner'

described previously •

, ~8

5.3.L Estimation of TotaL

If the sampling is carried out in two stages, the current total

is

T =aNr:

i=l

If the sampling is carried out in four stages, the current total is

= N Ni :s.j ~ijkaT ' r: r: 2,; &.. ci'iJ'kJ

i=l j=l k=l .8=1

As in Chapter 4, four types of estimators which utilize past as

well as present information from the constituent samples can be used.

Since the theory can be generalized to multi-stage sampling of any

degree as indicated in 5.2 , we will illustrate here only for the two-

stage case.

5.3.L1. A Linear Composite Estimator. The structure of the

linear composite estimator for the ath occasion is

~

where a-lT = the linear composite estimator of IT,a-

A

Ta,a-l

is the unbiased estimator of aT based on those (l-~)n first-stage

units which occur from order (a-l)~n+l to n+(a-2)~n, and common to

the (a_l)th occasion,

A 1 n+(a-2)~n Ni m a-1Yij

a-l,aT

= (l-~)n i=(a-f)~n+l Pi j~l m

•

J.

]29

is the unbiased estimator of a_1T based on the same (l-~)n first

stage units used on the ath occasion,

c1ijm

is the unbiased estimator of aT based on the set of n first-stage

units which occur from order (a-l)~n+l to~+(a-l)~n. This estimator

would be used to estimate T if the sampling is done only for onea

occasion, or if the sampler does not wish to utilize past information

from the previous occasions.

~ :~.~

Expe ct e. d val u eo f aT. ~ Intuitively, we would expect~

aT to be an unbiased estimator of aT. For example, when a = 3 !.~.,

when the sampling is carried out for 3 successive occasions, we will

have

(5.18)

and

~

E(3T)

~

Since E(2T) = 2T as shown in Chapter 4, we will have

A

E(3T) = Q[2T + 3T - 2T] + (l-Q) 3T .= 3T •

This will be true for a = 4, 5 and so on, so that it will be

generally true.

~

Variance of aT. From (5.14), 'we have

130

When the number of occasions ~ is not so large, for example, when

a =3 we will have

It is necessary to determine what may be termed, the sub-variances~

and sub-covariances which make up the total variance of 3T •~

Var (2T) is given previously.

It can be shown that

var(3,2T) = Xl-~)n (3Vy) , (where 3Vy= 3a~ + 3~)

= ~ P eYi _ T)2 + ~ ~ (3c{) (Ni-m)i=l i Pi 3 i=l Pi m (Ni-l)

131

Also,

where

N.(2Yij - 2Yi)(3Yij - 3Yi) (Ni-m)J. 1E - (N. -1)

,j=l N. m

J. J.

.... 1Var ( T) =- ( V ) ,

3. n 3 Y

"'.... 1COY (2,3T'3T) =n (2.3Vy) ,and

'" '" 1COY (3,2T, 3T) =n (3VY)

~ '" ~Now, to find the expressions for COY (2T, 3,2T) , COY (2T'g,3T) ,

*COY (2~'3T), we proceed as follows:

COY (2~)' 3,2T) = COY [{Q(lT + 2,lT -1,2T) + (1~)2T} , 3,2T]

= Q[COV (IT, 3,2T) + COY (2,lT, 3,2T)

- COY (1,2T , 3,2T)] + (l-Q) COY (2'r'3,2T) •

The following diagram shows the overlapping parts of the first-stage

samples on each of the three occasions. An inspection of this diagram will

help in the determination of covariances.

132

order

n+tJ,nn

o

./'--2,3Tr .....

I '"I T -,,- ............

Slon '"2nd 2T

occasion '" I3rd 2,1~ 3

TI

occasi n --- --'

1

1stoccs

It can be shown that

for

= 0

COY (IT'3,2T) =nfi:~)ln [1.3Vy]1

for 2" S Il <1

where

~i (lYij - 1~i)(3Yij- 3Yi)j=l Ni

1 (Ni - m)

m(Ni

- 1)

Next

1= 0 for 2" S Il < 1

where

'\

~i (2Yij - 2Yi)(3Yij - 3~i) 1j=l Ni m

(N. -m),~

133

for

1= 0 for 2 S ~ < 1 ,

and

1 [ J- v .n 2.3 y

Hence

Cov (2~' 3,.2T.) • Q[~( V) + (1-2~) ( V). ~ 1.3 Y (1_~)2n 2.3 Y

_ (1-2~) ( )J ( ) 1 () 12 1 3v + 1-Q - r'. 3V for 0 < ~ < 2- ,

(1-~) n· y n ~. Y

1= (l-Q) - ( V)n 2.3 y1

for 2' S ~ < 1.

The same procedure when applied to Cov (2~' 2,lT) and Cov (2~' 3T)yields:

~ ....Cov (2~' 2,3T) •

and

1= (l-Q) - ( V )2 2 Y

for

-~ ( V)J + (l-Q) (1-~) ( V), for 0 < ~ <-21~ 1.3 Y n 2.3 Y

= (1~) (1-~) ( V) f 1 < < 1-, n 2.3 Y or 2' - ~

and,

134

Combining these results, we will have from (5.20)

+ 2Q [~1-2» (~) + (1~2fl) (2.3Vy) _ (1-2fl) (1.3Vy)}

1-fl n (1_fl)2 n (1_fl)2 n

+ 2(1-Q) (2.3Vy ) _ 2Q { ~1-2» (1.2VY ) + (1-2fl) (2Vy )n 1-fl n (1_fl)2 n

_ (J.-2fl) (1.2Vy

)} _ 2(1-Q) ! ( V ) _ 2 ( V)](1_fl)2 n n 2 Y (l-fl)n 2.3 Y

+ (1_Q)2 ! ( V ) + 2Q(1-Q) [Q (1-2fl) ( V)n 3 Y t n 1.3 Y

+ ~1-2» ( V) _~ ( V)t + (l-Q) (l-fl) ( V)1-fl n 2.3 Y ~ 1.3 Y) n 2.3 Y

Var ( ~) = Q2 [Q2 [-E...} 1Vy + {1 + Q2 ..k.} 2Vy + {2Qfl2 _ 2Qf.l3 1-fl n l 1-fl n

V V V V_ 2Q2fl2} llx- + --lL.. +~ + 2(1-Q) (2.3 Y)

(l-fl)n (l-fl)n (l-fl)n n

V V V_ 2(l-Q) (U) _ 2 [~ + (1_Q)2 (U) +

n (l-fl)ru n

+ 2'1(1-'1) [(1-'1) (1;;) (2.3Vy) + ~ (3Vy) - ~ (2.3Vy )] ,

1for 2' =s fl < 1 (5.21)

1351 1

For example where I.l = 2' and Q = 2' ' we will have

Var ( ~) = ...l... (1VY) +...2... (2VY) + 20 (-D.)3 16 n 16 n Ib n

Under the assumptions that

l v = 2V = 3V = V ,Y Y Y Y

and

V - V1.2 Y - 2.3 y ,

also defining

Vp = 1.2 Y

-V1VY -v:}";we have

=V2.3 Y ,

If Var (3~) is compared with Var (3T), the variance of the

simple linear estimator

A 1 ni2l.ln Ni m 3Yij

3T = - L: -p L: ,

n i=2l.ln+l i j=l m

we will find that

(5.24 )

The percent gain in efficiency of the linear COIDP9site estimator

x 100 •

136

The percent gain in efficiency is tabulated in Table 5.1.

~ "Table 5.1. Percent gain in efficiency of 3T over 3T for

1 1f.l=2' Q=2


.6 negative

·7 4.57

.8 21.21

·9 44.14

.95 59.20

~

Comparing the nature of the gain in efficiency of 3T with that of~. 1. 1

2T for Q = 2' f.l =2·' we see that the gain is much higher. This is

because information from both the first and the second occasions is used.-- ---If the saIIWler wants to use only the information from t.he second occa-

~

sion, the form of the estimator will be like 2T and the percent gain

in efficiency will be as in Table 4.1.

When the proportion of first-stage units partially replaced is

1 ~less than 2"' a more cOIIWlicated expression of Var (3T) will be in-

volved (see the first expression in (5.21».

For example, whenJ.l. = ~ and Q:: ~ , we have

...2... (1.3VY) ~ (~)- 48 n - 48 n (5.25 )

...

137~

To compare the efficiency of 3T with that of the simple estimator

3T for this case, we will again make the following assumptions:

(i) V = V = V = Vly 2y 3Y Y

and

(iii)

V1.3 Y11/

=V

2·3 Y ,

Under such assumptions, we will have

V.1L

n

*The percent gain in efficiency of 3T over

lOp2 + 5!tpl - 36

132 - 54Pl - 10P2x 100

We tabulate the percent gain in efficiency for a series of assumed

values of Pl and P2 in Table 5.2.

11/ 2- In practice, we would expect that P2 < Pl' or perhaps P2 ii Pl;

an ass'UlTlPtion similar to the latter one has been made by many authors.

-Table 5.2. *Percent gain in efficiency of 3T over

assumed values of Pl and P2, for ~

'"3T for some

= ~, Q =~

138

...

..

2 %Gain in efficiencyPl P2=Pl

.60 .36 0.00

·70 .49 7·50

.80 .64 16.50

.90 .81 26.16

·95 .90 33·89

Comparing with Table 4.1, we see that if the correlation pattern

is approximately as assumed, the use of the linear composite estimator

*3T which utilizes past information from~ the first and second

occasions will again be more efficient than the one based on the

second occasion alone.

The case when a = 4 or 5 can be treated similarly. However, when

the number of occasions (!.!., a) becomes larger and larger, the exact

*expression for Var (aT ) given by (5 .19) becomes too involved. An

approximate expression was obtained involving many assumptions on

variance and covariance stability. Discussion on this ~oint is omitted

in view of its algebraic complexity.

5.3.1.2. A Modified Linear Composite Estimator. The structure

of the modified linear composite estimator when the sampling is done

for a > 2 successive occasions is

"-where Tex,ex-l"-

and Tare as defined in (5.15) and (5.16)ex-l,ex

139

.. "- 1 n+(ex-l)~n Ni mT=- L: - L:

ex, ex ~n i=n+(ex-2 )~n+l Pi j=l

..

is the unbiased estimator of exT based on the ~n first-stage units~,

which occur from order n+(ex-2)~n+l to n+(ex-l)~n. ex_1T is the

modified linear composite estimator of ex_1T •~

Expected value of exT'. It can be shown by an argument similar to~ 1:::

that used in the case of exT that exT' is also an unbiased estimator

of exT •~

Variance of T'. From (5.26), we haveex

It can be shown that the three covariances in the last square

bracket are zero, so that

• 140

For example, when a = 3 we have

*Now, Var (2T') has been given previously.

(5.29 )

1IJ.n

Cov (T T) - "T::'""l~_. 3,2' 2,3 - (l-lJ.)n

III A

To obtain expressions for Cov (2T', 3,2T)

we proceed as follows:

,..", "" ,.."Cov (IT, 3,2T) , Cov (2,lT, 3,2T) , Cov (1,2T, 3,2T)

have been derived previously.

From the diagrain showing the overlapping parts of the first-stage

samples on the three occasions given below

141

order

A

T

,occasion

I2,2'"~

sion "-2nd 2,3T -

occasionI"3rd 2 IT

"-

1,2T~ .-A--_.......

1stocca

we see by inspection that

With the above results, we obtain

1for 0 < ~ < 2 '

Similarly,

1.2Vy] + (l-Q) rl-~)n (2Vy)'

for 0 < ~ < ~

1= (l-Q) ( )

l-~n

1for""2 :s ~ < 1 .

142

Using all these results in (5.29), we have

1for 0 < J..I. < 2" '

and

{

V V V V+ 2Q2 1.2 Y _ U _ U-L} + (1_Q)2 (~) + 1 (V)

n n {l-J..I.)n j.ln (l-J..I.)n 3 Y

For example, when 1 1J..I. = 2" ' Q = 2' we will have

Under the assumption that

and

v - V and defining1.2 y - 2.3 y ,

P =

we will have

V1.2 y =v2.3 Y ,

6 [ 2. 10 1 VVar (3T') = 1 + Ib - Ib PJ + .*If T'

3'" '"is compared to the simple estimator 3T where 3T is as

defined previously, we will have the relation:

The percent gain in efficiency of

( ) _ (10p-5)G12 P - 21-10p x 100 •

*T'3

over '"3T for this case is

The gain in efficiency ::i.s tabulated in Table 5.3.

Table 5.3. Percent gain in efficiency of1 1

J..l=2"' Q=2"

"over 3T for

P

.6

·7.8

·9·95

%Gain in efficiency

6.6614.29

23.07

33·33

39·13

• 144*, *,Comparing the gain in efficiency of 3T with 2T for this

case (see Table 4.7) we see again that the gain in efficiency is in-

creased when we use past information from both the first and second

occasions.

1 *When J.l. < 2' J a more complicated expression for Var (3T') is

involved (see the first expression for Var (3~') in (5.30». For the

1 1case J.l. = 3" J Q = 2' J we have

V V V VVar (3~f) = -l (l..1:) + 2.. (U) + 2.£ (U)_ ..£.. (1.3 Y)

32 n 32.n 32 n 32 n

Under the assumptions that

V = V = V =Vly 2y 3y y

and

V = V1.2 Y 2.3 y

and by writing

we have

• 145

~ [40 - ~~l - 2P2]

VVar (3T') = ..L.

n

[1 +8 - (6Pl + 2P2 )] v

= ..L.32 . n

and

* 1 1which implies that 3T' (for ~ =3 ' Q =2) is less efficient thanA ;$l,

the simple estimator 3T as in case of 2~ •

To make the estimator 3~' worthwhile, a proper choice of Q must

be made. For example, if we use Q =t keeping ~ =~ , we will have

for this case,

Under the assumptions given above, we will have

_ A + {286 - 37.8pl - 108P2} V- Var (3T) 512 +

The percent gain in efficiency of ~,3

for this case is

378pl + 108P2 - 286G13(P) = [798 - 378pl - 108P2] x 100.

The gain for some values of Pl and P2 is tabulated in Table

•Table 5.4.

APercent gain in efficiency of 3Tf

1.1. =~ , Q =t for some values of

for

146

2 %Gain in efficiencyPl P2=Pl

.6 .36 negative

·7 .49 6·56

.8 .64 20.04

.9 .81 38.25

.95 ·90 49.69

5.3.1.3. A General Linear Estimator. A general linear estimator

which can be used to estimate aT, when a > 2 is

"'T* - ( T"'*) + b ( T) + ( T) + d( T)a - a a-l a-l,a c a,a-l a,a (5.34 )

where a-lT* is an unbiased estimator of a-lT and is of the same

* '" A Atype as aT; a-l,aT, a,a-1T and a,rxT are as previously defined.

"'*As in the case of 2T, we find that we must choose the weights in

(5.34) such that

b = -a

and

d = l-c

for aT* to be an unbiased estimator of aT. Using these values we find

.."'*Further, by minimizing Var (aT ) with respect to a and c

simultaneously, we obtain the best values of the weights, viz.

147

a = [[var (a,aT)J[Cov (a,a-1T, a-1,aT) - Cov (a-1T*, a,a-1T)~ /

~var (a,aT) + Var (a,a-1T)] [Var (a_1T*) + Var (a-1,aT)

"'* '" A* A ) (A* A)]~- 2 Cov (a_IT, a-l,aT)] - [cov(a_lT , a,a-1T - Cov a_1T 'a-l,aT J

(5.36 )

and

o =[[var (a,aT)][var (a-1T*) + Var (a-1,i) - 2 Cov (a-1T*, a-1,aT)~ /

~var (a,aT) + Var (a,a_1T)][var (a_1T*) + Var (a-1,aT)

- 2 Cov (a_1T*, a-1,aT)] - [Cov (a_1T*, a,a-1T) - Cov (a-1T*'a-1,aT)]~

and consequently (for ready reference)

(1-0) = [[var (a,a-1T)l [Var (a-1T*) + Var (a-1,i) - 2 Cov(a_1T*'a_1,aT)]

- [Cov (a-1T*, a,a-1T) - Cov (a-1T*, a-1,aT)]~ / [[var (a,aT)

+ Var (a,a_1T)][var (a_1T*) + Var (a-l,aT) - 2 Cov (a_1T*, a-l,aT)]

- [Cov (a_1T*, a,a-1T) - Cov (a_1T*, a-1,aT)]~ .

(5.38)

With these values of a, c and l-c substituted in (5.35), we will

obtain the formula for aT*.. The expression however is omtted in view

of its length.

,

148A* A A A

Noting that Cov (a_1T , a,aT), Cov (a-l,aT, a,aT) andA A

Cov (a,a-1T, a,aT) are each equal to zero, we find variance of the

estimator aT* to be

where a and c are as defined in (5.36) and (5.37).

For example, when a =3, we will have

A A* A A* A

+ Var (3,2T)][var (2T ) + Var (2,3T) - 2 Cov (2T , 2,3T)]

- [Cov (2T*, 3,2T) - Cov (2T*, 2,3T)]'

c =[[var (3,3T)][var (2T*) + Var (2,3T) - 2 Cov (2T*, 2'3T)~/

[[var (3,3T) + Var (3,2T)][var (2T*) + Var (2,3T) - 2 COv (2T*'2,3T)]

"'* "'* A 21- [Cov (2T , 3,2@) - Cov (2T , 2,3T)] J ."'* A "'* '"All expressions except Cov (2T , 3,2T) and Cov (2T , 2,3T) have been

derived previously. To find the above two covariances, ·we proceed as

follows:

149

From (4.66) we have

(1-1J. ) A A

+ Cov (2,lT, 3,2T)(1_1J.2p2 )

1J.(1-lJ.p2 ) A A

+ Cov (2,2T, 3,2T) •(1_1J.2p2 )

~ ".With the aid of the diagram used in the discussion of 3T and 3T,

we find that

and

for ~:S IJ. < 1

for 0 < IJ. < ~ ,

150

and

Under the assumptions that V = V =V and V = V2 y 3 y y 2.3 Y 1.2 y ,

so that

V= _...,;;;1;,..;.2;;;....M.Y _

-V1Vy -V2Vy

1we will have for example, when j..L = 2'

,..

a =

and

c = 8

,

The variance of the estimator 3T* given by (5.39) is found to

be

A* AThe percent gain in efficiency of 3T over 3T as measured by

is

[Var (3T) - Var (3T*)]

Var (3T*)x 100

~ ~l

G14(P) = [(64 - l6p2 ) - [(4-2p2)(p-l)} 2]2 - 128(4-p2)(4-3p2)

- 2 [32 - 8p2 - {(4_2p2 )(p-l~ 2J x 100 / 128(4-p2 )(4-3p2)

+ 2 ~2 - 8p2 - t(4_2p2 )(p-l)}2J 2

The gain in efficiency for given values of p is tabulated in

Table 5.5.

Table 5.5. Percent gain in efficiency of /r* over /r for 1-1. = ~

..

/, %Gain in efficiencyp

.6 11.61

·7 16.76

.8 23·83

·9 34.11

·95 41.19

Comparing the nature of the gain in efficiency using this type

of estimator to the case where a = 2, we see again that there is a

slight increase in the gain in efficiency when the sampling is done on

three successive occasions (see Table 4.13).

1When 1-1. < 2' ' the expression for the weights a and c and

Var (3T*) will be more involved and will not be given here.

When a > 3, the problem can be treated similarly but the algebra

will become heavier as the number of occasions (!.~., a) increases.

152

5.3.1.4. A Ratio-Type Composite Estimator. The ratio-type

composite estimator which may also be used to estimate aT when a > 2

is

'"+ (l-Q) Ta

~*where a_1T is the ratio-type composite estimator of a_1T !.~.,

the total of population characteristic of interest for the (a_l)th

'" ."" '" '"occasion. lT, rv-l fVT, rvT , Q are as defined previously_a ,a- .....- ,..... .....~*The estimator aT is not an unbiased estimator of aT but when

properly used, may yield a more efficient estimator than the simple

'"estimator aT •

~*Variance of aT. From (5.40), we have

Var

Applying Lemma II and III, and using only the leading terms, we

~*obtain the approximate expression for Var (aT ) as

~* A **Assuming that ECa_1T) ~ a~lT C!.~.J the bias of a_1T , as an

estimate of IT, is relatively small) the approximate expression fora-**Vax CaT) is reduced to

.. 154

The expressions tor covariances such as

be worked out by applying Lemma III.

For example, when <l = .3 , we will have

(5.44)

'"All expressions except for Cov (3,2T,

llt* '"Cov (2T , 3T), have been given previously.

155llt* '" "'*

2T ), Cov (2,3T, 2T ),

To obtain the approximate

•

expressions for these covariances, we proceed as follows:

Consider

'"COy (2~' 3,2T) ~ COY [Q (2,1;)(1T)} + (l-Q) 2T, 3,2T]

1,2'"

2 IT '" '" '"= Q Cov t(~)(lT), 3,2T} + (l-Q) Cov (2T, 3,2T).

1,2T

By applying Lemma III, we will have

2T -Except for the ratio ("'T) which is a constant, the four sub-covariances

1 ~in the expression above are as previously given in the case Of

3T.

Similarly,

(5.4q)

•

..

156

and

.2'2.2. Estimation of Change in Total

To estimate the change in totals between the current and previous

occasions, !.~., aT - a_1T , where a > 2, four types of estimators

which are extensions of the case a = 2, can be used. We will

indicate the nature of the extension of the theory only briefly.

5.2 .2.1. The Estimator Based on the Linear Composite Estimator.

The estimator based on the linear composite estimator is

~ ~ ~

D = T - Ta-l a a-l

~ Awhere aT and a_1T are the linear composite estimators defined

previously. In view of the demonstration given in connection with

(5.18) it is unbiased.-

Variance: From (5.48), we have

(5.49)

The variances and covariances on the R.H.S. of (5.49) for specific

cases can be worked out as in the equation leading to (5.21).

5.3.2.2. The Estimator Based on the Modified Linear Composite

Estimator. The structure of this estimator is

~ ~ A.u' = T' - T'a-l a a-l

A A,where aT' and a-lT

as defined in (5.26).

157

(5 .50)

are the modified linear composite estimators

Variance: From (5.50), we have

(5.51)

The variances and covariances on the R.H.S. can also be worked out as

in the equation leading to (5.30).

5.3.2.3. A General Linear Estimator. This estimator is

By imposing the condition that

a '+ b' = -1

and

C ' + d' - 1- ,""*so that Da _l takes the form

shall be an unbiased estimator of

(5 .53)

""* I IFurther, minimizing Var (Da

_l ) with respect to a and c, yields

a 1 = [{Var (a,a.T) + COY (a-l,aT'a.,a-lT)} {COV (a.-l,aT'a,a.-lT)

"* ""} { ""- COV(a_lT 'a.,a-lT) - Var (a-l,aT)

- Cov (a-1T*'a-1,aT)} {var (a,a-1T) + Var(a)}J / [{Var (a-1T*)

and

- COY (a.~l~ 'a-l,aT)} {Cov(a._l,a.T'a,a_lT)

cov(a._lT:'a,a-lT)}] I [{var(a._l~) + var(a._l,a.T)

- 2 cov(a_lT~'a_l,aT)}{var(a.,a_lT) + var(a,aT)}

- {cov<CX_1,aT'a,a_1T) - cov(a_1T*'a,a-1'1!)}'

Variance. From (5.53) the variance of the estimator ~-l is

Var (B~_l) = a 12 var(a_lT*) + (l+a 1 )2var(a_l,a~) + c ,2Var (a,a_l;)

" (A* A)+ 2a c Cov a-lT_'a,a-lT

159

5.3.2.4. The Estimator Based on the Ratio-Type Estimator. This

estimator which is not an unbiased estimator of aT - a_1T is given by

A A

A * ~where aT and a_1T are the ratio~type composite estimators of aT

and a_1T respectively.

Variance. From (5.57), we will have

each component

A A*= var(a~) + var(a_l~_)

on the R.H.S. of (5.58) can be worked out for specific

cases as in the equation leading to (5.44).

160

6. SUMMARY AND CONCLUSIONS

6.1. SUlIIllI.8.ry

A multi-stage sampling design and the resulting estimation theory

particularly intended for large scale sample surveys on successive

occasions is developed. The sam;pling design is kept general in the

sense that the selection of the first-stage sampling units is done with

arbitrary (unequal) probabilities. A technique of partial replacement

of first-stage sampling units based on their order of occurrence in

the preliminary saIllJ?le is proposed. This technique is intended to

serve two purposes:

(i) To spread the burden of reporting among respondents which can

be expected to minimize response resistance.

(ii) To enable the sampler to utilize the correlation over time in

the reduction of the variance of several estimates of population totals

and change in totals.

The estimation theory is presented. Four types of estimator which

can be used to estimate the totals of the population characteristic of

interest and the changes in such totals are discussed. Of the four,

the first three estimators which are referred to in this thesis as

the linearcoIllJ?osite estimator~ the modified linear cOIllJ?osite estimator

and the general linear estimator are unbiased estimators •. The fourth

estimator which is referred to as the ratio-type cOIllJ?osite estimator is

a biased estimator but this bias is likely to be small. The expressions

for the variances of these estimators are given. The per cent gains in

.1

162

the Appendix, the estimation of variances and covariances in the rele

vant variance formulae will always be sim;ple in view of the mutual

statistical independence between the first-stage units brought about

by the sim;ple ex;pedient of sam;pling with replacement after each draw

in forming the preliminary sample.

• 16;

7 • LIST OF REFERENCES

1. Cochran, W. G. 196; • Sampling Techniques, 2nd Edition. JohnWiley and Sons, New York.

2. Des Raj. 1954. On sampling with varying probabilities in multistage designs. Ganita, 5:45..;51.

3 . Des Raj. 1965. On sampling over two occasions with probabilityproportionate to size. Annals of Mathematical Statistics,;6:327-3;0.

4. Eckler, A. R. 1955. Rotation sampling. Annals of MathematicalStatistics, ;6:664-685.

5. God.aIribe, V. P. 1955. A unified theory of sampling from fin,itepopulation. Journal of the Royal Statistical Society, SeriesB, 17:269-278.

6. Hansen, M. H., Hurwitz, W. N., Nisselson, H. and Steinberg, J.1955 • The redesign of the census current population survey'.Journal of the American Statistical Association, 50:701-719.

•

7· Hansen, M. H., Hurwitz, W. N., and Madow, W. G.Survey Methods and Theory, Vol. I and II.Sons, New York •

1953 • SampleJohn Wiley and

8. Jessen, R. J. 1942. Statistical investigation of a sample surveyfor obtaining farm facts. Iowa Agricultural Experiment StationResearch Bulletin, No. 304. Ames, Iowa.

9. Koop, J. c. 1963. On the axioms of sample formation and theirbearing on the construction of linear estimators in samplingtheory for finite universes. Metrika 7 (2 and 3): 81-114and 165-204.

10. Madow, W. G. 1949. On the theory of systematic sampling, II.Annals of Mathematical Statistics, 20:333-;54.

11. Onate, B. T. 1960. Development of multi-stage designs forstatistical surveys in the Philippines. M1meo-Multi1ithSeries, No. ;, Statistical Laboratory, Iowa State University,Ames, Iowa.

12 • Patterson, H. D. 1950. Sampling on successive occasions withpartial replacement of units. Journal of the Royal StatisticalSociety, Series B, 12:241-255.

..

164

13 . Rao, J. N. K. 1961. On sampling with varying probabilities andwith replacement in sub-sampling designs. Journal of· theIndian Society of Agricultural Statistics, 13:211-217.

14. Rao, J. N. K. and Graham, J. E. 1964. Rotation designs forsampling on repeated occasions. Journal of the AmericanStatistical Association, 59:492-509.

15. Sukhatme, P. V. 1954. Sampling Theory for Surveys with Applications. The Indian Society of Agricultural Statistics, NewDelhi, India, and the Iowa State College Press, Ames, Iowa.

16. Tikkiwal, B. D. 1955. Multiphase sampling on successive occasions. Unpublished Ph. D. thesis, Department of ExperimentalStatistics, North Carolina State University, Raleigh, NorthCarolina.

17. Tikkiwal, B. D. 1958. Theory of successive two-stage sampling.Abstract in Annals of Mathematical Statistics, 29:1291.

18. Ware, K. D. and Cunia, T. 1962. Continuous forest inventory withpartial replacement of samples. Forest Science, Monograph 3.The Society of American Foresters.

19. Woodruff, R. S. 1959. The use of rotating samples in the CensusBureau's monthly surveys. Proc. Social Statistics Section,American Statistical Association, 130-138.

20. Yates, F. 1960. Sampling Method for Censuses and Surveys.Charles Griffin and Co., London.

..8. APP~DICES

8.1. Note on the Estimation of Sa.rtW11ng Variances

To verify that 1(j~y + 1(j~ is unbiasedly est11!lated by

165

where

nL:

1=1

y'(1 1 ,,)2-- T

Pi 1

n-1,

and

m

1Yi' = N. L:J. j=l

Consider

... 1 n N1 mT=- L: L:

1 n i=l Pi j=l

nL:

1=1

n-11 n 1Yi' 2 2

= - E[ L: (-) - n ")n-1 i=l Pi 1

T

n Y' 2= 1: L: E(l i) - 1rr?- =

n 1=1 Pi

166

Now

N ~2

10"i= Z

Pi-

i=lm

N ~2

·lO"i= Z

Pii=lm

• •

.. Similarly" it can be verified that 20"~y + 20"~ is unbiasedly esti

mated by

and

n~n

Ei=J..m+l n-l where

161

And similarly, it can be verified that 1.2CTbyy + 1.2CTwyy is

unbiasedly estimated by

nE

'i=;.tn+l (l-IJ )n-l

'"2,lT)

8.20 Note on the Efficiency of the Estimators when lV ~ 2VY . Y

The nature of the gt:l.in in efficiency using the estimators discussed

in Chapter 4 has been examined under the assumption of equal varianceso

When the variances differ on each occasion, we would expect that the

gain in efficiency will be different from what has been tabulated.

We will indicate the nature of the change by the following exam;ples:

~I. The Linear Composite Estimator 2T

1 1For example, Q "" 2' ' IJ == 2" and assuming that lVy "" A. 2Vy

when A. > 0 and ~ 1-

From (4.35), we will have,

where

Table 8.2. '" '"The percent gain in efficiency of 2T over 2T when1 1 5Q=- IJ.=- and V =T.'(V)2' 2 ly Lj- 2y

p %Gain in Efficiency

.6 negative

·7 2.56

.8 12.04

·9 23.84

·95 30·55

169

Comparing with Table 8.1, it is interesting to note that, in

'"using this type of estimator (i.e. 2~)' the efficiency is increased

when lVy > 2Vy' otherwise it is decreased, value by value of p.

~f

II. The Modified Linear Com,posite Estimator 2T

For exgmple, Q = ~ IJ. = ~ and assuming again that lVy = A(2Vy)

where A > 0 and ~ 1-

From (4.53), we 'Will have by substitution

where

'" 2V

Var ( T) = U .:2 n

~I '"The per cent gain in efficiency of 2T over 2T is

[

2p Ii - A J.4 + A - 2p VA J x 100 •

NORTH CAROLINA STATE UNIVERSITY

INSTITUTE OF STATISTICS

(Mimeo Series available for distribution at cost)

423. Miller, H. D. Generalization of a theorem of Marcinkiewicz. 1965.424. Johnson, N. L. Paths and chains of random straight-line segment. 1965.425. Nasoetion, Andi H. An evaluation of two procedures to estimate genetic and environmental parameters in a simul·

taneous selfing and partial diallel test-crossing design. 1965.426. Bose, R. C. Error detecting and error correcting indexing systems for large serial numbers. 1965.427. Chakravarti, I. M. On the construction of different sets and their use in the search for orthogonal Latin squares and

error correcting codes. 1965.428. Bose, R. C. and Chakravarti, I. M. Hermitian varieties in a finite projective space PG (N, q"). 1965.429. Potthoff, R. F. and Maurice Whittinghill. Maximum-likelihood estimation of the proportion of non-paternity. 1965.430. Tranquilli, G. B. On the normality of independent random variables implied by intrinsic graph independence without

residues. 1965.431. Miller, H. D. Geometric ergodicity in a class of denumerable Markov chains. 1965.432. Braaten, M. O. The union of partial diallel mating designs and incomplete block environmental designs. 1965.433. Harvey, J. R. Fractional moments of a quadratic form in non-central normal random variables. 1965.434. Nixon, D. E. A study of the distributions of two test statistics for periodicity in variance. 1965.435. Basu, D. Problems related to the existence of maximal and minimal elements in some families of statistics (sub-fields).

1965.436. Gun, Atindramohan. The use of a preliminary test for interactions in the estimation of factorial means. 1965.437. Smith, W. L. A theorem on functions of characteristic functions and its application to some renewal theoretic random

walk problems. 1965.438. Nuri, Walid, Fourier methods in the study of variance fluctuations in time series analysis. 1965.439. Barlotti, A. Some topics in finite geometrical structures. 1965.440. Bhapkar, V. P. and Gary Koch. On the hypothesis of "no interaction" in the three dimensional contingency tables. 1965.441. Hall, W. J. Methods of sequentially testing composite hypotheses with special reference to the two-sample problem.

1965.442. Bohrer, R. E. On Bayes sequential design of experiments. Ph.D. Thesis. 1965.443. Hoeffding, Wassily. On probabilities of large deviations. 1965.444. Kanofsky, Paul B. Parametric confidence bands on cumulative distribution functions. 1965.445. Potthoff, R. F. A non-parametric test of whether two simple regression lines are parallel. 1965.446. Metzler, C. M., Gennard Matrone and H. L. Lucas, Jr. Estimation of transport rates by radioisotope studies of non-

steady-state systems. Ph.D. Thesis. 1965.447. Koch, Gary. A general approach to estimation of variance components. 1965.448. Assakul, Kwanchai. Testing hypotheses with categorical data subject to misclassification. Ph.D. Thesis. 1965.449. Bhapkar, V. P. and Gary Koch. On the hypothesis of "no interaction" in the four dimensional contingency tables. 1965.450. Bhapkar, V. P. Categorical data analogs of some multivariate tests. 1965.451. Chakravarti, I. M. Bounds on error correcting codes (non-random). 1965.452. Anderson, R. L. Non-balanced experimental designs for estimating variance components. Presented at Seminar on

Sampling of Bulk Materials, Tokyo, Japan. November 15-18, 1965.453. Quade, Dana. On analysis of variance for the k-sample problem. 1965.454. Ikeda, Sadao, JunjiroOgawa and Motoyasu Ogasawara. On the asymptotic distribution of F-statistics under the null

hypothesis in a randomized PBIB design and M associate classes under the Neyman model. December 1965.455. Ikeda, Sadao. On certain types of asymptotic equivalence of real probability distributions. I. Definitions and some of

their properties. December 1965.456. Yandle, David. A test of significance for comparing two different systems of stratifying the same population. Ph.D.

Thesis. 1965.457. Mesner, Dale M. The block structure of certain PBIB designs of partial geometric types. 1966.458. Sen, P. K. A class of permutation tests for stochastic independence. I. 1966.459. Koch, Gary. Some considerations about the effect of redundancies and restrictions in the general linear regression

model. 1966.460. van der Vaart, H. R. An elementary derivation of the Jordan normal form with an appendix on linear spaces. A

didactical report. 1966.461. Smith, Walter S. Some peculiar semi-Markov processes. 1966.462. Sen, P. K. On a class of non-parametric tests for bivariate interchangeability. 1966.

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