1 Inferences about a Mean Vector Shyh-Kang Jeng Department of Electrical Engineering/ Graduate...

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:,

ff

ff

ff

F

fFf

F

f

fff

ff

Ff

Ff

fF

ff

1111

FF-Distribution-Distribution

1212

Nature of Nature of TT22-Distribution-Distribution

pnppp

pnpp

np

n

jjj

Fpn

pnF

p

p

NWn

N

T

Xnn

XXXX

XnT

,1 d.f.,

1,

1

21,

0

1

10

2

)(

)1(

1 d.f.

d.f. calculus,by

),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


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


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


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


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


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


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


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


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


)-(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


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


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


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


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

n

jj

p

Fpnnm

pnnm

n

nmnm

n

n

nn

nm

nN

n

mN

XXSXX

ΣXX

XXXXX

Σ0XX

XXX

Σμ

8181


1,

12

21

1

)(

:'1

),0(:1

)(:1

pnnmp

jj

pj

p,nm-nn

Fpnnm

pnnmn

nmT

Nn

nm

Wn

XXSXX

ΣXX

SSSS

8282

EM AlgorithmEM AlgorithmPrediction stepPrediction step– Given some estimate of the unknown Given some estimate of the unknown

parameters, predict the contribution of parameters, predict the contribution of the missing observations to the the missing observations to the sufficient statistics sufficient statistics

Estimation stepEstimation step– Use the predicted statistics to compute Use the predicted statistics to compute

a revised estimate of the parametersa revised estimate of the parameters

Cycle from one step to the otherCycle from one step to the other

8383

Complete-Data Sufficient StatisticsComplete-Data Sufficient Statistics

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8484

Prediction Step for Prediction Step for Multivariate Normal DistributionMultivariate Normal Distribution

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of components available:

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8686

Estimation Step for Estimation Step for Multivariate Normal DistributionMultivariate Normal Distribution

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8787

Example 5.13: EM AlgorithmExample 5.13: EM Algorithm

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8888

Example 5.13: Prediction StepExample 5.13: Prediction Step

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9191

Example 5.13: Estimation StepExample 5.13: Estimation Step

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9292

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9393

Coverage Probability of Coverage Probability of the 95% Confidence Ellipsoidthe 95% Confidence Ellipsoid

1 Inferences about a Mean Vector Shyh-Kang Jeng Department of Electrical Engineering/ Graduate...

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