1 Inferences about a Mean Vector Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking.
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11
Inferences about a Mean VectorInferences about a Mean Vector
Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/Graduate Institute of Communication/Graduate Institute of Communication/
Graduate Institute of Networking and MultiGraduate Institute of Networking and Multimediamedia
22
InferenceInference
Reaching valid conclusions Reaching valid conclusions concerning a population on the basis concerning a population on the basis of information from a sampleof information from a sample
33
Plausibility of Plausibility of 00 as a Value for a as a Value for a
Normal Population MeanNormal Population Mean
n
ii
n
ii
n
XXn
sXn
Xns
Xt
XXX
H
H
1
22
1
0
21
01
00
1
1,
1,
/
:statistics test eAppropriat
population
normal a from sample Random:,,,
: hypothesis ealternativ sided)-(Two
: hypothesis Null
44
Student’s Student’s tt-distribution-distribution
2
12
2
1)
2(
)2
1(
)(
/
f
f
tf
f
f
tf
f
Zt
55
Student’s Student’s tt-distribution-distribution
66
Test of HypothesisTest of Hypothesis
)2/()()(
,i.e.
)2/(/
if level cesignificanat
offavor in Reject
210
120
2
10
10
n
n
txsxnt
tns
xt
HH
77
Confidence IntervalConfidence Interval
n
stx
n
stx
n
stx
%-α
tns
xH
nn
n
n
)2/()2/(
)2/(
interval confidence )1(100 in the lies
)2/(/
or levelat reject not Do
101
1
0
10
0
88
Plausibility of Plausibility of 00 as a Multivariate as a Multivariate
Normal Population MeanNormal Population Mean
n
jjj
n
jj
n
nn
n
nT
T
H
H
11
01
0
0
1
02
2
21
1
0
'1
1,
1
'
'
:statistics sHotelling'
population
normal a from sample Random:,,,
: hypothesis ealternativ sided)-(Two
: hypothesis Null
XXXXSXX
μXSμX
μXS
μX
XXX
μμ
μμ
0
0
99
TT22 as an as an FF-Distribution-Distribution
pnpF
pn
pnT
,
2 1:
1010
FF-Distribution-Distribution
21
21
11
,
2/
2
1
122
2
1
21
21
222
121
2122
21
:
122
2)(
0,/
/
lyrespective, and d.f. with t,independen:,
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ff
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f
fff
ff
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FF-Distribution-Distribution
1212
Nature of Nature of TT22-Distribution-Distribution
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np
n
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p
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NWn
N
T
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XXXX
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,1 d.f.,
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),0()(1
1)',0(
vectorrandom
normal temultivaria
d.f.
matrix
randomWishart
' vectorrandom
normal temultivaria
1
'
'
1313
Test of HypothesisTest of Hypothesis
)()1(
)()'(
if level cesignificanat
offavor in Reject
,
12
10
pnpFpn
pn
nT
HH
00 μxSμx
1414
Example 5.1 Evaluating Example 5.1 Evaluating TT22
1,223,22
2
1
0
4)23(
2)13(:
9
7
56
98
27/49/1
9/13/156983
27/49/1
9/13/1
93
34,
6
8
5
9,
38
610
96
FFT
T
S
SxμX
1515
Example 5.2 Testing aExample 5.2 Testing aMean VectorMean Vector
level %10 at the Reject 18.874.9
18.8)1.0(17
319)10.0(
)(
)1( : valueCritical
74.9,
402.0002.0258.0
002.0006.0022.0
258.0022.0586.0
628.3640.5810.1
640.5788.199010.10
810.1010.10879.2
,
965.9
400.45
640.4
normalitycheck ,20 0.10. level aat Test
10504':,10504':
02
17,3,
21
10
HT
FFpn
pn
T
n
HH
pnp
S
Sx
μμ
1616
Invariance of Invariance of TT22-Statistic-Statistic
0
10
01
0
0,1
0,2
00,
1
)'(
'')'(
'
,)(
''1
1,
singular-non : ,
μxSμx
μxCCSCCμx
μySμy
dCμμdCμYμ
CSCyyyySdxCy
CdCXY
YyY
YY
y
n
n
nT
E
n
n
jjj
1717
TT22-Statistic from -Statistic from Likelihood Ratio TestLikelihood Ratio Test
),(max
),(max ratio Likelihood
'2
1exp
2
1
),(
1ˆ,'
1ˆ
ˆ2
1),(max
,
0
10
102/2/
0
11
2/2/2/,
Σμ
Σμ
μxΣμxΣ
Σμ
xxμxxxxΣ
ΣΣμ
Σμ
Σ
Σμ
L
L
L
nn
eL
n
jjjnnp
n
jj
n
jjj
npnnp
1818
TT22-Statistic from -Statistic from Likelihood Ratio TestLikelihood Ratio Test
n
jjj
npnnp
n
jjj
n
jjj
n
jjj
n
eL
1000
2/2/
02/
0
100
1
100
1
10
10
'1ˆ
ˆ2
1),(max
'tr
'tr'
μxμxΣ
ΣΣμ
μxμxΣ
μxμxΣμxΣμx
Σ
1919
Result 4.10Result 4.10
BΣ
Σ
BΣ
B
BΣ
b
ebe
b
pp
pp
bppb
bb
2/1for only holding
equality with , definite positive allfor
211
scalar positive :
matrix definite positive symmetric:
)(
2/)tr( 1
2020
Likelihood Ratio TestLikelihood Ratio Test
c
H
HH
n
n
jjj
n
jjj
n
n
2/
100
1
2/
0
0
0100
0/2
'
'
ˆ
ˆ if
level at the Reject
:against :
of test ratio Likelihood
lambda Wilks':ˆ/ˆ
μxμx
xxxx
Σ
Σ
μμμμ
ΣΣ
2121
Result 5.1Result 5.1
12/2
0100
2
21
11
because : vs.:
of test likelihood the toequivalent is test
),( from sample random:,,,
n
T
HH
T
N
n
pn
μμμμ
ΣμXXX
2222
Proof of Result 5.1Proof of Result 5.1
121
11212211211
22121122
2221
1211
0
01
|
|
1|
|'
AAAAAAAAAAA
AA
AA
μx
μxxxxx
A
n
nn
jjj
2323
Proof of Result 5.1Proof of Result 5.1
12
0
/2
2
0
10000
1
0
1
10
1
100
11
ˆ
ˆ
11ˆˆ
'''
''1'
'')1(
n
T
n
Tnn
n
n
n
n
n
jjj
n
jjj
n
jjj
n
jjj
n
jjj
Σ
Σ
ΣΣ
μxμxμxμxxxxx
μxxxxxμxxxxx
μxμxxxxx
2424
Computing Computing TT22 from Determinants from Determinants
)1(
'
')1(
)1(ˆ
ˆ)1(
1
100
02
n
n
nn
T
n
jjj
n
jjj
xxxx
μxμx
Σ
Σ
2525
General Likelihood Ratio MethodGeneral Likelihood Ratio Method
cL
L
HH
H
L
)(max
)(max
if : offavor in Rejects
:
sample randomby function likelihood:)(
,parameters populationunknown :
0
010
00
θ
θ
Θθ
Θθ
θ
Θθθ
Θθ
Θθ
2626
Result 5.2Result 5.2
00
2
ofdimension ofdimension -
where variable,random aely approximat is
)(max
)(maxln2ln2
large, is size samplewhen
0
0
ΘΘ
θ
θ
Θθ
Θθ
L
LΛ-
n
2727
100(1100(1--))%% Confidence Region Confidence Region
level cesignificanat
offavor in :reject not test will
thefor which all of consistingRegion
1 truecover the will)(
where)R(:region confidence )%-100(1
array databy
determined valueslikely ofregion :)(
,parameters populationunknown :
1
00
0
H
H
RP
R
θθ
θ
θX
X
X
θX
Θθθ
2828
100(1100(1--))%% Confidence Region Confidence Region
)()1(
'
such that allby determined ellipsoid The
)(
of interval The
,1
21
12
pnp
n
Fpn
npn
txsxn
μxSμx
μ
:Case Normal teMultivaria
:Case Normal Univariate
2929
Axes of the Confidence EllipsoidAxes of the Confidence Ellipsoid
iii
ipnpi Fpnn
np
eSe
e
x
where
)()(
)1(
are axes the,center at the beginning
,
3030
Example 5.3 :Example 5.3 :Microwave Oven RadiationMicrowave Oven Radiation
704.0710.0,002.0
710.0704.0,026.0
228.200391.163
391.163018.203
0146.00117.0
0117.00144.0,
603.0
564.0
opendoor with radiation measured
closeddoor with radiation measured
'22
'11
1
42
41
e
e
S
Sx
x
x
3131
Example 5.3 :Example 5.3 :95% Confidence Region95% Confidence Region
0.05 level cesignifican at the 589.0
562.0: offavor in
rejected benot would589.0
562.0: test By this
region. confidence 95% in the is
62.630.1
589.0603.0
562.0564.0
228.200391.163
391.163018.203589.0603.0562.0564.042
589.0562.0'
62.6)05.0(40
)41(2
603.0
564.0
228.200391.163
391.163018.203603.0564.042
1
0
40,2
2
121
μ
μ
μ
μ
H
H
F
3232
Example 5.3 : Example 5.3 : 95% Confidence Ellipse for 95% Confidence Ellipse for
018.0
)23.3()40(42
)41(2002.0)(
)(
)1(
064.0
)23.3()40(42
)41(2026.0)(
)(
)1(
:axesminor -semi andmajor -semi
603.0564.0'
:center
,2
,1
pnp
pnp
Fpnn
np
Fpnn
np
x
3333
Example 5.3 : Example 5.3 : 95% Confidence Ellipse for 95% Confidence Ellipse for
3434
Simultaneous Simultaneous Confidence StatementsConfidence Statements
Sometimes we need confidence Sometimes we need confidence statements about the individual statements about the individual component meanscomponent means
All if the separate confidence All if the separate confidence statements should hold statements should hold simultaneously with a specified high simultaneously with a specified high probabilityprobability
3535
Concept of Simultaneously Concept of Simultaneously Confidence StatementsConfidence Statements
3636
Confidence Interval of Linear Confidence Interval of Linear Combination of VariablesCombination of Variables
nt
nt
tt
n
ns
zt
sz
NZ
ZN
nn
n
z
Z
z
zZ
p
Saaxaμa
Saaxa
Saa
μaxa
Saaxa
ΣaaμaΣaaμa
XaΣμX
')(''
')('
)(
'
)''(
/
','
)','(:,','
'),,(:
11
21
2
2
3737
Maximum Maximum tt22 Value for All Value for All aa
μxS
a
μxSμx
Saa
μxa
Saa
μxa
Saa
μxa
aa
1
21
22
22
toalproportion for occurs maximum
'
'
'max
'
'max
'
'maxmax
Tn
nn
nt
aa
3838
Maximization LemmaMaximization Lemma
dBd
Bxx
dx
Bxx
dBdBxxdx
dBx
dBdBxx
dx
dB
x
12
12
1
12
0
''
'
0'
)')('('
:Proof
0for when attained maximum
''
'max
orgiven vect matrix, definite positive
cc
3939
Result 5.3: Result 5.3: TT22 Interval Interval
1least at y probabilit with 'contain will
')(
)1(
and ')(
)1(
by determined
interval) ( interval the, allfor usly Simultaneo
),( from sample random:,,,
,
,
2
21
μa
Saaxa
Saaxa
a
ΣμXXX
pnp
pnp
pn
Fpnn
np'
Fpnn
np'
T
N
4040
Comparison of Comparison of tt- and - and TT22-Intervals-Intervals
4141
Simultaneous Simultaneous TT22-Intervals-Intervals
n
sF
pn
npx
n
sF
pn
npx
n
sF
pn
npx
n
sF
pn
npx
n
sF
pn
npx
n
sF
pn
npx
pppnppp
pppnpp
pnppnp
pnppnp
)()(
)1()(
)(
)1(
)()(
)1()(
)(
)1(
)()(
)1()(
)(
)1(
,,
22,22
22,2
11,11
11,1
4242
Example 5.4: Shadows of the Example 5.4: Shadows of the Confidence EllipsoidConfidence Ellipsoid
4343
Example 5.5Example 5.5
11.2337.2325.217
37.2305.12651.600
25.21751.60034.5691
,
13.25
69.54
59.526
87
sciencefor score CQT:
for verbal score CQT:
history and science socialfor score CLEP:
3
2
1
Sx
n
X
X
X
4444
Example 5.5Example 5.5
32
233322,32
32
321
1
,,
for interval confidence 95%least -atan is 3.1229.56 i.e.,
2)05.0(
)(
)1(
are interval confidence its of points end ,for ]1,1,0['
61.2665.23,16.5822.51,88.54930.50387
34.569129.859.526
87
34.569129.859.526
29.8)05.0(387
)187(3)(
)1(
n
sssF
pn
npxx
FFpn
np
pnp
pnppnp
a
4545
Example 5.5: Confidence Ellipses Example 5.5: Confidence Ellipses for Pairs of Meansfor Pairs of Means
4646
One-at-a-Time IntervalsOne-at-a-Time Intervals
n
stx
n
stx
n
stx
n
stx
n
stx
n
stx
ppnpp
ppnp
nn
nn
)2/()2/(
)2/()2/(
)2/()2/(
11
22122
2212
11111
1111
4747
Bonferroni InequalityBonferroni Inequality
m
m
ii
m
ii
ii
ii
ii
CPCP
CPCP
miCP
aC
21
11
'
1
true11false 1
false oneleast at 1 true all
,,2,1,1true
about statement confidence:
4848
Bonferroni Method of Bonferroni Method of Multiple ComparisonsMultiple Comparisons
n
s
ptx
n
s
ptx
n
s
ptx
n
s
ptx
mmm
in
s
mtxP
pmm
ppnpp
ppnp
nn
m
iii
ni
i
2
2
2
2
11
all, contains 2
,/
11
11111
1111
terms
1
4949
Example 5.6Example 5.6
646.0560.0or 42
0146.0327.2603.00125.0
607.0521.0or 42
0144.0327.2564.00125.0
327.2)2
025.0(
025.02/05.0,2
2
22412
1
11411
41
n
stx
n
stx
t
p i
5050
Example 5.6Example 5.6
5151
(Length of Bonferroni Interval )/(Length of Bonferroni Interval )/(Length of (Length of TT22-Interval)-Interval)
5252
Limit Distribution of Limit Distribution of Statistical DistanceStatistical Distance
n-p
n
n
n-p
n
pnn
N
p
p
p
largefor
ely approximat:)()'(
large is
y whenprobabilithigh with toclose
largefor
ely approximat :)()'(
size sample largefor )1
,(nearly :
21
21
μXSμX
ΣS
μXΣμX
ΣμX
5353
Result 5.4Result 5.4
20
10
0100
21
'
if ,ely approximat cesignifican of level aat
,: offavor in rejected is :
large
covariance definite positive and mean with
population a from sample random:,,,
p
n
n
HH
pn
μxSμx
μμμμ
Σμ
XXX
5454
Result 5.5Result 5.5
1ely approximat
yprobabilit with ,every for ,'contain will
')('
large
covariance definite positive and mean with
population a from sample random:,,,
2
21
aμa
Saax
Σμ
XXX
na
pn
p
n
5555
Result 5.5Result 5.5
)-(1 confidence with ,contain
)(
, pairs allfor
)( )(
)( )(
statements confidence ussimultaneo )%-100(1
2
1
22
11211
1121
ki
pkk
ii
kkik
ikiikkii
ki
ppppp
pppp
pp
x
x
ss
ssxxn
n
sx
n
sx
n
sx
n
sx
5656
Example 5.7: Musical Aptitude Example 5.7: Musical Aptitude Profile for 96 Finish StudentsProfile for 96 Finish Students
5757
Example 5.7: Simultaneous 90% Example 5.7: Simultaneous 90% Confidence LimitsConfidence Limits
plausiblenot are componentsmeter tempo,melody,
22222331342731
studentsAmerican of Profile
13.2427.21
39.2361.20,93.2427.22
01.3639.32,75.3605.34
67.2853.24,14.3006.26
02.12)10.0(,)10.0(
'0
7
65
43
21
27
27
n
sx ii
i
5858
One-at-a-Time and Bonferroni ConfOne-at-a-Time and Bonferroni Confidence Intervalsidence Intervals
n
s
pzx
n
s
pzx
n
szx
n
szx
iiii
iii
iiii
iii
22
intervals confidence Bonferroni
22
intervals confidence time-a-at-One
5959
Large-Sample 95% Intervals for Large-Sample 95% Intervals for Example 5.7Example 5.7
6060
95% Intervals for Example 5.795% Intervals for Example 5.7
6161
Control ChartControl Chart
Represents collected data to Represents collected data to evaluate the capabilities and stability evaluate the capabilities and stability of the processof the process
Identify occurrences of special Identify occurrences of special causes of variation that come from causes of variation that come from outside of the usual processoutside of the usual process
6262
Example 5.8: Overtime Hours for Example 5.8: Overtime Hours for a Police Departmenta Police Department
6363
Example 5.8 Example 5.8 Univariate Control ChartUnivariate Control Chart
6464
Monitoring a Sample for StabilityMonitoring a Sample for Stability
ondistributi
square-chi a as )()'( eApproximat
oft independennot isbut normal, is
1)(
0)(
),(
as ddistributetly independen:,,,
1
21
XXSXX
SXX
ΣXXCov
XX
Σμ
XXX
jj
j
j
j
p
n
n
n
E
N
6565
Example 5.9: Example 5.9: 99% Ellipse Format Chart99% Ellipse Format Chart
6666
Example 5.9: -Chart for Example 5.9: -Chart for XX22X
6767
Example 5.10: Example 5.10: TT22 Chart Chart for for XX11 and and XX22
6868
Example 5.11: Robotic WeldersExample 5.11: Robotic Welders
bleeach variaon nsobservatio
successivefor n correlatio serial eappreciabl No
reasonable is assumption Normal
(cfm) flow Gas (inert):
(in/min) speed Feed:
(amps)Current :
(volts) Voltage:
4
3
2
1
X
X
X
X
6969
Example 5.11: Example 5.11: TT22 Chart Chart
7070
Example 5.11: 99% Quality Control Example 5.11: 99% Quality Control Ellipse for ln(Gas flow) and voltageEllipse for ln(Gas flow) and voltage
7171
Example 5.11: -Chart for Example 5.11: -Chart for ln(Gas flow)ln(Gas flow)
X
7272
Control Regions for Future Control Regions for Future Individual ObservationsIndividual Observations
Set for future observations from Set for future observations from collected data when process is stablecollected data when process is stable
Forecast or prediction regionForecast or prediction region– in which a future observation is in which a future observation is
expected to lieexpected to lie
7373
Result 5.6Result 5.6
pnp
pn
Fpn
pnn
nT
N
,
12
21
)1( as ddistribute is
'1
ondistributi same the
fromn observatio future:
),( astly independen:,,,
XXSXX
X
ΣμXXX
7474
Proof of Result 5.6Proof of Result 5.6
pnp
npp
Fpn
pn
n
n
WNn
n
n
nn
E
,1
1,
)1(:'
1
)(:),,0(:1
1
1)()()(
0)(
XXSXX
ΣSΣXX
Σ
ΣΣXCovXCovXXCov
XX
7575
Result 4.8Result 4.8
n
jj
n
jj
nn
n
j
n
jjjjpnn
jp
n
b
c
bbb
ccNccc
N
1
2
1
2
122112
1 1
222111
j
21
)()'(
)'()(
matrix covariancewith
normaljoint are and
)(,:
),(:
tindependenmutually :,,,
ΣΣcb
ΣcbΣ
VXXXV
ΣμXXXV
ΣμX
XXX
7676
Example 5.12 Control EllipseExample 5.12 Control Ellipse
7777
TT22-Chart for Future Observations-Chart for Future Observations
)05.0()(
)1(UCL
0 LCL
order in time
'1
Plot
,
12
pnpFpn
pn
n
nT
xxSxx
7878
Control Chart Based on Control Chart Based on Subsample MeansSubsample Means
ΣΣ
XX
XXXXX
XX
ΣXX
XXX
Σμ
nm
n
mn
n
n
n
n
n
nnnnn
nm
nN
nj
mN
j
njjj
j
pj
n
jjj
p
)1(1111
Cov1
Cov1
1
1111)
11(Cov
)Cov(
))1(
,0(:
1, at timemean subsample :
timesame at the sampled be units 1),,( :Process
2
2
12
2
111
1
7979
Control Chart Based on Control Chart Based on Subsample MeansSubsample Means
1,
12
21
1
)(
:'1
),0(:1
)(:1
pnnmp
jj
pj
p,nm-nn
Fpnnm
pnnmn
nmT
Nn
nm
Wn
XXSXX
ΣXX
SSSS
8080
Control Regions for Future SubsamControl Regions for Future Subsample Observationsple Observations
1,1
1
1
1
)1(
)(:'
1
1Cov
1Cov
11Cov)Cov()Cov(
))1(
,(:
1,mean subsample future :
timesame at the sampled be units 1),,( :Process
pnnmp
n
pj
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