Tail probability approximation for one and two sample U- statistics by the saddlepoint method

155 Pages·1993·4.4 MB·English

by  Sharma, Divakar, 1963-

TAILPROBABILITYAPPROXIMATIONFORONEANDTWOSAMPLE
U-STATISTICSBYTHESADDLEPOINTMETHOD
By
DIVAKARSHARMA
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1993
Copyright©1993
by
DivakarSharma
ToMyWife,
Supriya
ACKNOWLEDGEMENTS
Iwouldliketoacknowledgeallthosehiddencontributorswhohavebeena
constantsourceofinspirationandshownunlimitedsupportduringmyresearch. I
wishtoexpressmysincereappreciationandthanks
• tomyadvisor,Dr.MyronS. Chang,forhisadvice, guidanceandconstructive
criticismandforprovidingaconstantstimulustoimprovethequalityofresearch;
• tomyfavoritelong-termcompanion,Supriya,forsharingwithmeallthepain
andfrustrationexperiencedduringthisperiod,forbeingcompletelysupportive
andunderstandingandaboveallforbeingaconstantsourceofencouragement;
• toDr.M.Ghosh,Dr.P.V.Rao,Dr.J.ShusterandDr.M.Raoforproviding
suggestionstofurtherimprovethemanuscript;
• tomyparents,PranandRadha,forexposingmetoallbasicoperationsofhuman
functioningandmakingmestrongenoughtosurvive;tomybrother,LaUt,for
remindingmewhyIshouldalwaysstrivetobebetter;
• toGailandLaurieformakingitpossibletobringthismanuscriptintoprint;
• andfinallytoallmyfriendsandcolleaguesin CBARDSatMerckResearch
LaboratoriesforthesupportIreceivedincompletingmyresearchanddegree.
IV
TABLEOFCONTENTS
Page
ACKNOWLEDGEMENTS
iii
LISTOFTABLES vi
LISTOFFIGURES x
ABSTRACT xii
CHAPTERS
1 INTRODUCTION 1
1.1 Review 1
1.2 Problem 2
2 SADDLEPOINTAPPROXIMATIONS 6
3 ONESAMPLE: U-STATISTIC 15
3.1 NotationandResults 16
3.2 U-Statistic:DegreeK 26
3.3 Example 36
4 TWOSAMPLE: U-STATISTICS 49
4.1 TwoSample: Degree(1,1) 50
4.2 TwoSample: Degree(Kj,K2) 64
4.3 Example 77
5 COMPARISONOFTHETAILPROBABILITYAPPROXIMATION
BYTHEMETHODOFEDGEWORTHEXPANSIONAND
SADDLEPOINTAPPROXIMATION 88
6 POWEROFONESAMPLEU-STATISTICS 97
6.1 DiscussionandMethods 97
6.2 Results 103
V
7 POWER; TWOSAMPLEU-STATISTICS 118
7.1 DiscussionandMethods 118
7.2 Results 125
REFERENCES 138
BIOGRAPHICALSKETCH 140
VI
LISTOFTABLES
Page
3.1 Comparisonoftailprobabilityapproximationof
Wilcoxonsignedrankteststatistic(N=17)between
exact(null)distribution,normalapproximationand
saddlepointapproximation 44
4.1 Comparisonoftailprobabilityapproximationof
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
betweenexact(null)distribution,normal
approximationandsaddlepointapproximations 82
5.1 Comparisonoftailprobabilityapproximationof
Wilcoxonsignedrankteststatistic(N=17)between
exact(null)distribution,normalapproximation,
Edgeworthexpansionandsaddlepointapproximation 93
6.1 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionforpower
calculationofWilcoxonSignedRanktestunder
normalshiftalternatives 104
6.2 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionforpower
calculationofWilcoxonSignedRanktestunder
logisticshiftalternatives 105
6.3 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionforpower
calculationofWilcoxonSignedRanktestunder
Cauchyshiftalternatives 105
6.4 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
normalalternative(A=0.253347)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 107
vii
6.5 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
normalalternative(A=0.524401)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 108
6.6 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
normalalternative(A=0.841621)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 109
6.7 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
logisticalternative(A=0.405465)betweenexact
distribution,normalapproximationandsaddlepoint
no
approximations
6.8 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
logisticalternative(A=0.847298)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 111
6.9 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
logisticalternative(A=1.38629)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 112
6.10 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
Cauchyalternative(A=0.32492)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 113
6.11 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
Cauchyalternative(A=0.726543)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 114
6.12 Comparisonoftailprobabilityapproximationof
WilcoxonSignedRankteststatistic(N=17)under
Cauchyalternative(A=1.37638)betweenexact
viu
distribution,normalapproximationandsaddlepoint
approximations 115
7.1 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionfornormalshift
alternatives 123
7.2 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionforlogisticshift
alternatives 124
7.3 Computationofexpectationsneededtoapproximate
thecumulantgeneratingfunctionforCauchyshift
alternatives 124
7.4 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
undernormalalternative(A=0.253347)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 126
7.5 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
undernormalalternative(A=0.524401)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 127
7.6 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
undernormalalternative(A=0.841621)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 128
7.7 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underlogisticalternative(A=0.405465)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 129
7.8 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underlogisticalternative(A=0.847298)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 130
IX
7.9 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underlogisticalternative(A=1.38629)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 131
7.10 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underCauchyalternative(A=0.32492)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 132
7.11 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underCauchyalternative(A=0.726543)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 133
7.12 Comparisonoftailprobabilityapproximationfor
Wilcoxon-Mann-Whitneyteststatistic(n=5,m=10)
underCauchyalternative(A=1.37638)betweenexact
distribution,normalapproximationandsaddlepoint
approximations 134
X