1 Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression Raghu Nandan Sengupta Department of Industrial and.

1

Use of Asymmetric Loss Functions in Sequential Estimation Problem for the

Multiple Linear Regression

Raghu Nandan SenguptaDepartment of Industrial and Management Engineering

Indian Institute of Technology Kanpur

R.N.Sengupta, IME Dept., IIT Kanpur, INDIA 2

What is this talk all about?

Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression

Raghu Nandan SenguptaJOURNAL OF APPLIED STATISTICS (2008) , 35, 245-261

Abstract: While estimating in a practical situation, asymmetric loss functions are preferred over squared error loss functions, as the former is more appropriate than the latter in many estimation problems. We consider here the problem of fixed precision point estimation of a linear parametric function in beta for the multiple linear regression model using asymmetric loss functions. Due to the presence of nuissance parameters, the sample size for the estimation problem is not known before hand and hence we take the recourse of adaptive multistage sampling methodologies. We discuss here some multistage sampling techniques and compare the performances of these methodologies using simulation runs. The implementation of the codes for our proposed models is accomplished utilizing MATLAB 7.0.1 program run on a Pentium IV machine. Finally we highlight the significance of such asymmetric loss functions with few practical examples.

Key words and phrases: loss function; risk; bounded risk; asymmetric loss function; LINEX loss function; relative LINEX loss function; stopping rule; multistage sampling procedure; purely sequential sampling procedure; batch sequential sampling procedure


Background

Any population is characterized by X (random variable) which has a particular distribution given by its cumulative distribution function (cdf), where the cdf is given by

P[X x] = F(x ; )Note:

1) In general we select a sample {X1, X2,….., Xn} of random observations to estimate

2) The statistics is given by Tn = T(X1, X2,….., Xn) which is an estimator of

3) If we consider as the error in our estimation process, then = (Tn - )


Loss functions

, the error results in different types of LOSS FUNCTIONS, denoted by L() and few examples are

1) Absolute error loss function

L() = ||2) Squared error loss (SEL) function

L() = 2

3) Linear exponential (LINEX) loss function, [Zellner (1986)], a type of asymmetric loss function

L() = b{ea - a -1}

where a ( 0) is the shape parameter and b (>0) is the scale parameter

4) Balanced loss function (BLF), [Zellner (1994)]

L() = w{g() – g(Tn)}/{g() – g(Tn)} + (1-w)(Tn - )2

where 0w1


LINEX loss function

L() = b{ea - a -1}

L() L()

a > 0 a < 0


Few examples where asymmetric loss functions can be used

Marketing strategyExponential survival model, where X is the life time of a component with a pdf

f(x;,) = (1/)exp{-(x - )/}where = minimum guarantee/warranty time/period1/ = failure rate

Note: Use a LINEX loss function with an appropriatevalue for a



Construction of a dam

Underestimation of height of dam is more serious than overestimation

Note: Use a LINEX loss function with a (< 0)



Reliability of equipments

Exponential life time of equipments, where X is the life of an equipment with a pdf

f(x;) = (1/)exp{-x/}such that the reliability function R(t) is given by

R(t) = P[X > t] = exp{-t/}Over estimation of the reliability function can have

marked consequence than under estimation

Note: Use a LINEX loss function with a (> 0)


Risk

RISK = EXPECTED LOSS

Let us consider a simple example

1) Consider we choose X1, X2,….., Xn (i.i.d) from X ~ N(,2), but with both and 2 unknown.

2) We are interested in estimating using

3) The loss function for the LINEX loss is of the form

4) The risk is given by

, , 1na Xn n nR X E L X E e a X

1

1 n

n ii

X Xn

, 1na Xn nL X e a X


Concept of Bounded Risk

RISK w (given or a known quantity)

5) The risk is given by

6) For bounded risk we must have

7) As both and 2 are unknown we take the recourse of Sequential Sampling Techniques

2 2

, exp 12na

R Xn

2 2

2log 1e

an

w


Different Sequential Sampling Methodologies

1) Two stage sampling procedure [Stein (1945)]

2) Purely sequential sampling procedure [Ray (1957)]

3) Three stage sampling procedure [Hall (1981), Mukhopadhyay (1980)]

4) Accelerated sequential sampling procedure [Hall (1983), Mukhopadhyay and Solanky (1991)]

5) Batch sequential sampling procedure [Liu (1997)]


Estimation problem for the multiple linear regression

In the context of the multiple linear regression problem formulation we have just discussed for the first paper we now deal with the problem of estimation considering LINEX loss function ~~

/ l



Given n data points the usual least square error (LSE) estimator of and the forecasted value of are

n = (Xn Xn )–1XnYn

/

~ ~n nl



However, Zellner(1986) has shown that when 2 is known, under LINEX loss, the estimator thus found for SEL is inadmissible, being dominated (in terms of risk) by the estimator

2 1* / / /

~ ~ ~~ 2n n n na

l l X X l



Theorem: Under LINEX loss, the estimator

is dominated by the estimator of the form

even when 2 is unknown

Here we replace 2 by it usual predictor 2n

n

2 1** / /

~ ~2n

n n n na

l X X l



Thus under asymmetric loss function the shrinkage estimators and dominates

The corresponding risk of the shrinkage estimatoris given by

when 2 is known

*n n

*n

2 2 1/ /

~ ~2 n na

l X X l

**n



Now if the corresponding risk is bounded then we have

But if 2 is unknown we have to solve the problem of finding the optimal sample size by taking the recourse of some adaptive sampling methodologies about which we have discussed before

2 2 1/ /

~ ~2 n na

l X X l w



Purely sequential sampling procedure

m m+1 N-1 N

2 2 1/ /

~ ~inf :

2n

n na

N n m l X X l w




One sampling stops we consider the two forecasted values

2 1** / /

~ ~2N

N N NNa

l X X l

/

~ ~N Nl




The corresponding risks are

, 1NaN NR E e a

**

** **, 1NaN NR E e a



Batch sequential sampling procedure

1) Choose a positive integer k and consider 0 < 1 < 2 < …< k < 1, thus the objective is to estimate k fractions of the sample size using sequential type sampling, but taking batches of observations at each stage.

2) We specify decreasing batch sizes for these k sampling stages as r1 > r2 >…> rk > 1.

3) In the final stage, sampling is done purely sequentially4) We start with an initial sample of size m and then, for t = 1,2,…,

define

m (m+r1*t1) (N-1*tk) N




1)

2)..

k-1)

k)

2 2 1/ /

1 1 1~ ~

inf :2n

n na

R n m r t m l X X l w

2 2 1/ /

2 1 2 1 2~ ~

inf :2n

n na

R n R r t R l X X l w

2 2 1/ /

1 1~ ~

inf :2n

k k k k k n na

R n R r t R l X X l w

2 2 1/ /

1~ ~

inf :2n

k k k n na

N n R r t R l X X l w




One sampling stops we consider the two forecasted values

2 1** / /

~ ~2N

N N NNa

l X X l

/

~ ~N Nl




The corresponding risks are

, 1NaN NR E e a

**

** **, 1NaN NR E e a



If the parameter value is very small then considering a relative LINEX loss function would be more practical and advisable than a LINEX loss function. The relative LINEX loss function and its corresponding risk is

L(,T) = ea(T/ -1) - a(T/ - 1) -1

R(,T) = E[L(,T)] = E[ea(T/ - 1) - a(T/ - 1) -1]



Similar bounded risk problem formulation for the estimated value was undertaken and corresponding sequential sampling methodologies were considered for the case of relative LINEX loss function


Simulation for the estimation problem for the multiple linear regression

Data set used for simulation

The manuscript describing the data can be found at www.spatial-statistics.com. One can refer Kelley and Barry (1997) for further details

The MLR is of the form

)()()()(log 33

2210 MIMIMIMHVe

)(log)(log)(log 654 PB

PTRMA eee

)(log)(log 87 HHP

ee

http://www.spatial-statistics.com/



Data set used for simulation

Where

1) MHV = Median house value

2) MI = Median income

3) MA = Housing median age

4) TR = Total rooms

5) B = Total bedrooms

6) P = Population

7) H = Households



For the LINEX loss function the following sampling methodologies were considered with the starting sample size m=10

PSL: Purely Sequential (m = 10)

BSL(1): Batch sequential (m = 10, k+1 = 3; 1 = 0.80, 2 = 0.90, r1 = 24, r2 = 16, r3 = 8)

BSL(2): Batch sequential (m = 10, k+1 = 3; 1 = 0.75, 2 = 0.85, r1 = 15, r2 = 10, r3 = 5)

For the relative LINEX loss function the following sampling methodologies were considered with the starting sample size m=10

PSRL: Purely Sequential (m = 10)

BSRL(1): Bath sequential (m = 10, k+1 = 3; 1 = 0.60, 2 = 0.90, r1 = 17, r2 = 13, r3 = 9)

BSRL(2): Batch sequential (m = 10, k+1 = 3; 1 = 0.70, 2 = 0.80, r1 = 10, r2 = 7, r3 = 4)



Consider a = -0.6 and w=0.03 for the relative LINEX loss function

%save

For BSRL(1)

42 4 0.020655 0.020421 -0.009788 -0.010013 84.60

For BSRL(2)

39 6 0.022570 0.022298 -0.007696 -0.007964 76.90

For PSRL

35 26 0.024785 0.024460 -0.005298 -0.005626 ----

N SO ),( NR **( , )NR )},({2 NRnd **2 { ( , )}ndNR


Acknowledgements for my visit IUSSTF Fellowship Foundation

Princeton University, USA

Prof. Jianqing Fan, ORFE Department, Princeton University, USA

Prof. Lawrence M. Seiford, IOE Department, University of Michigan, Ann Arbor

Prof. Katta G. Murty, IOE Department, University of Michigan, Ann Arbor

Prof. Romesh Saigal, IOE Department, University of Michigan, Ann Arbor


Contact Detail

Raghu Nandan Sengupta

Assistant Professor

Industrial & Management Engineering Department

Indian Institute of Technology Kanpur

Kanpur 208 016, UP, INDIA

Ph: +91-512-259-6607; Fax: +91-512-259-7553

Email: [email protected]

mailto:[email protected]


Research Areas

Sequential Analysis Financial Optimization Statistical Reliability Use of different Meta Heuristics

Techniques for Optimization

34

List of Publications

1. LINEX Loss Function and its Statistical Application – A Review ; (co-authors Saiobal Chattopadhyay and Ajit. Chaturvedi), DECISION , Jan-Dec, 1999, 26 , 1-4, 51-76.

2. Sequential Estimation of a Linear Function of Normal Means Under Asymmetric Loss Function ; (co-authors Saibal Chattopadhyay and Ajit Chaturvedi), METRIKA , 2000, 52 , 3, 225-235.

3. Asymmetric Penalized Prediction Using Adaptive Sampling Procedures; (co-authors Saibal Chattopadhyay and Sujay Datta), SEQUENTIAL ANALYSIS , 2005, 24 , 1, 23-43.

4. Three-Stage and Accelerated Sequential Point Estimation of the Normal Mean Using LINEX Loss Function; (co-author Saibal Chattopadhyay), STATISTICS , 2006, 40 , 1, 39-49.

R.N.Sengupta, IME Dept., IIT Kanpur, INDIA


List of Publications

5. Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression, JOURNAL OF APPLIED STATISTICS , 2008, 35 , 8, 245-261.

6. "Impact of information sharing and lead time on bullwhip effect and on-hand inventory" ; (co-authors, Sunil Agrawal and Kripa Shanker), EUROPEAN JOURNAL OF OPERATIONAL RESEARCH , (Accepted and forthcoming).

7. Bankruptcy Prediction using Artificial Immune Systems" , (co-author Rohit Singh), LECTURE NOTES IN COMPUTER SCIENCE (LNCS), L.N.de Castro, F.J.Zuben and H.Knidel (Eds.), 2007, 4628 , 131-141.


Thank you all

1 Use of Asymmetric Loss Functions in Sequential Estimation Problem for the Multiple Linear Regression Raghu Nandan Sengupta Department of Industrial and.

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