Probability and statistical models with applications

609 Pages·2000·5.132 MB·English

by  CH. A. Charalambides, M.V. Koutras, N. Balakrishnan

Preface
TheophilosCacoulloshasmadesignificantcontributionstomanyareas
ofprobabilityandstatisticsincludingDiscriminantAnalysis,Estimation
for Compound and Truncated Power Series Distributions, Characteriza-
tions, Differential Variance Bounds, Variational Type Inequalities, and
Limit Theorems. This is reflected in his lifetime publications list and
thenumerouscitationshispublicationshavereceivedoverthepastthree
decades.
Wehavebeenassociatedwithhimondifferentlevelsprofessionallyand
personally. We have benefited greatly from him on both these grounds.
We have enjoyed his poetry and his knowledge of history, and admired
his honesty, sincerity, and dedication.
This volume has been put together in order to (i) review some of the
recentdevelopmentsinStatisticalScience,(ii)highlightsomeofthenew
noteworthy results and illustrate their applications, and (iii) point out
possiblenewdirectionstopursue. Withthesegoalsinmind,anumberof
authorsactivelyinvolvedintheoreticalandappliedaspectsofStatistical
Science were invited to write an article for this volume. The articles so
collected have been carefully organized into this volume in the form of
38 chapters. For the convenience of the readers, the volume has been
divided into following six parts:
• APPROXIMATIONS, BOUNDS, AND INEQUALITIES
• PROBABILITY AND STOCHASTIC PROCESSES
• DISTRIBUTIONS, CHARACTERIZATIONS, AND
APPLICATIONS
• TIME SERIES, LINEAR, AND NON-LINEAR MODELS
• INFERENCE AND APPLICATIONS
• APPLICATIONS TO BIOLOGY AND MEDICINE
From the above list, it should be clear that recent advances in both
theoretical and applied aspects of Statistical Science have received due
attention in this volume. Most of the authors who have contributed to
thisvolumewerepresentattheconferenceinhonorofProfessorTheophi-
los Cacoullos that was organized by us at the University of Athens,
Greece during June 3–6, 1999. However, it should be stressed here that
this volume is not a proceedings of this conference, but rather a vol-
©2001 CRC Press LLC
©2001 CRC Press LLC
ume comprised of carefully collected articles with specific editorial goals
(mentioned earlier) in mind. The articles in this volume were refereed,
and we express our sincere thanks to all the referees for their diligent
work and commitment.
It has been a very pleasant experience corresponding with all the au-
thors involved, and it is with great pleasure that we dedicated this vol-
ume to Theophilos Cacoullos. We sincerely hope that this work will
be of interest to mathematicians, theoretical and applied statisticians,
and graduate students.
Our sincere thanks go to all the authors who have contributed to this
volume, and provided great support and cooperation throughout the
course of this project. Special thanks go to Mrs. Debbie Iscoe for the
excellent typesetting of the entire volume. Our final gratitude goes to
Mr. Robert Stern (Editor, CRC Press) for the invitation to take on this
project and providing constant encouragement.
Ch. A. Charalambides Athens, Greece
Markos V. Koutras Athens, Greece
N. Balakrishnan Hamilton, Canada
April 2000
©2001 CRC Press LLC
©2001 CRC Press LLC
Contents
Preface
ListofContributors
ListofTables
ListofFigures
TheophilosN.Cacoullos—AViewofhisCareer
PublicationsofTheophilosN.Cacoullos
TheTenCommandmentsforaStatistician
PartI.Approximation,Bounds,andInequalities
1NonuniformBoundsinProbabilityApproximations
UsingStein’sMethod
LouisH.Y.Chen
1.1Introduction
1.2PoissonApproximation
1.3BinomialApproximation:BinaryExpansion
ofaRandomInteger
1.4NormalApproximation
1.5Conclusion
References
2ProbabilityInequalitiesforMultivariateDistributions
withApplicationstoStatistics
JosephGlaz
2.1IntroductionandSummary
2.2PositiveDependenceandProduct-TypeInequalities
2.3NegativeDependenceandProduct-TypeInequalities
2.4Bonferroni-TypeInequalities
2.5Applications
2.5.1SequentialAnalysis
2.5.2ADiscreteScanStatistic
2.5.3AnApproximationforaMultinomial
Distribution
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2.5.4AConditionalDiscreteScanStatistic
2.5.5SimultaneousPredictioninTimeSeries
Models
References
3ApplicationsofCompoundPoissonApproximation
A.D.Barbour,O.Chryssaphinou,andE.Vaggelatou
3.1Introduction
3.2FirstApplications
3.2.1Runs
3.2.2SequenceMatching
3.3WordCounts
3.4DiscussionandNumericalExamples
References
4CompoundPoissonApproximationforSumsof
DependentRandomVariables
MichaelV.BoutsikasandMarkosV.Koutras
4.1Introduction
4.2PreliminariesandNotations
4.3MainResults
4.4ExamplesofApplications
4.4.1ACompoundPoissonApproximationfor
TruncatedMovingSumofi.i.d.r.v.s.
4.4.2TheNumberofOverlappingSuccessRuns
inaStationaryTwo-StateMarkovChain
References
5UnifiedVarianceBoundsandaStein-TypeIdentity
N.PapadatosandV.Papathanasiou
5.1Introduction
5.2PropertiesoftheTransformation
5.3ApplicationtoVarianceBounds
References
6ProbabilityInequalitiesforU-Statistics
TasosC.Christofides
6.1Introduction
6.2Preliminaries
6.3ProbabilityInequalities
References
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II.ProbabilityandStochasticProcesses
7TheoryandApplicationsofDecoupling
VictordelaPe˜ na andT.L.Lai
7.1CompleteDecouplingofMarginalLawsand
One-SidedBounds
7.2TangentSequencesandConditionally
IndependentVariables
7.3BasicDecouplingInequalitiesforTangent
Sequences
7.4ApplicationstoMartingaleInequalitiesand
ExponentialTailProbabilityBounds
7.5DecouplingofMultilinearForms,U-Statistics
andU-Processes
7.6TotalDecouplingofStoppingTimes
7.7PrincipleofConditioninginWeak
Convergence
7.8Conclusion
References
8ANoteontheProbabilityofRapidExtinction
ofAllelesin aWright-FisherProcess
F.Papangelou
8.1Introduction
8.2TheLowerBoundforBoundarySets
References
9StochasticIntegralFunctionalsinanAsymptotic
SplitStateSpace
V.S.KorolyukandN.Limnios
9.1Introduction
9.2Preliminaries
9.3PhaseMergingSchemeforMarkovJump
Processes
9.4AverageofStochasticIntegralFunctional
9.5DiffusionApproximationofStochastic
IntegralFunctional
9.5.1SingleSplittingStateSpace
9.5.2DoubleSplitStateSpace
9.6IntegralFunctionalwithPerturbedKernel
References
©2001 CRC Press LLC
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10BusyPeriodsforSomeQueueswithDeterministic
InterarrivalorServiceTimes
ClaudeLef`evreandPhilippePicard
10.1Introduction
10.2Preliminaries:ABasicClassofPolynomials
10.2.1ConstructionoftheBasicPolynomials
10.2.2AGeneralizedAppellStructure
10.3The D /M(Q)/1Queue
g
10.3.1ModelandNotation
10.3.2ExactDistributionof N
r
10.4The M(Q)/D /1Queue
g
10.4.1ModelandNotation
10.4.2ExactDistributionof N
r
References
11TheEvolutionofPopulationStructureofthe
PerturbedNon-HomogeneousSemi-Markov
System
P.-C.G.VassiliouandH.Tsakiridou
11.1Introduction
11.2ThePerturbedNon-Homogeneous
Semi-MarkovSystem
11.3TheExpectedPopulationStructurewith
RespecttotheFirstPassageTimeProbabilities
11.4TheExpectedPopulationStructurewith
RespecttotheDurationof aMembership
inaState
11.5TheExpectedPopulationStructurewith
RespecttotheStateOccupancyofaMembership
11.6TheExpectedPopulationStructurewith
RespecttotheCountingTransitionProbabilities
References
III.Distributions,Characterizations,andApplications
12CharacterizationsofSomeExponentialFamilies
BasedonSurvivalDistributionsandMoments
M.Albassam,C .R .Rao,andD.N.Shanbhag
12.1Introduction
12.2AnAuxiliaryLemma
12.3CharacterizationsBasedonSurvival
Distributions
12.4CharacterizationsBasedonMoments
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References
13BivariateDistributionsCompatibleorNearly
CompatiblewithGivenConditionalInformation
B.C.Arnold,E.Castillo,andJ.M.Sarabia
13.1Introduction
13.2ImpreciseSpecification
13.3PreciseSpecification
13.4AnExample
References
14ACharacterizationofaDistributionArisingfrom
AbsorptionSampling
AdrienneW.Kemp
14.1Introduction
14.2TheCharacterizationTheorem
14.3AnApplication
References
15RefinementsofInequalitiesforSymmetricFunctions
IngramOlkin
References
16GeneralOccupancyDistributions
Ch.A.Charalambides
16.1Introduction
16.2AGeneralRandomOccupancyModel
16.3SpecialOccupancyDistributions
16.3.1GeometricProbabilities
16.3.2BernoulliProbabilities
References
17ASkew t Distribution
M.C.Jones
17.1Introduction
17.2DerivationofSkew t Density
17.3PropertiesofSkew t Distribution
17.4AFirstBivariateSkew t Distribution
17.5ASecondBivariateSkew t Distribution
References
18OnthePosteriorMomentsforTruncationParameter
DistributionsandIdentifiabilitybyPosteriorMeanfor
ExponentialDistributionwithLocationParameters
Y.MaandN.Balakrishnan
18.1Introduction
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18.2PosteriorMoments
18.3Examples
18.4IdentifiabilitybyPosteriorMean
18.5AnIllustrativeExample
References
19DistributionsofRandomVolumeswithoutUsing
IntegralGeometryTechniques
A.M.Mathai
19.1Introduction
19.2EvaluationofArbitraryMomentsofthe
RandomVolumes
19.2.1Matrix-VariateDistributionsfor X
19.2.2Type-1BetaDistributionfor X
19.2.3TheCasewhentheRowsof X are
IndependentlyDistributed
19.2.4Type-1BetaDistributedIndependent
Rowsof X
19.2.5Type-2BetaDistributedIndependent
Rowsof X
19.2.6IndependentlyGaussianDistributed
Points
19.2.7Distributionsofthe r-Contents
References
IV.TimeSeries,Linear,andNon-LinearModels
20CointegrationofEconomicTimeSeries
T.W.Anderson
20.1Introduction
20.2RegressionModels
20.3SimultaneousEquationModels
20.4CanonicalAnalysisandtheReduced
RankRegressionEstimator
20.5AutoregressiveProcesses
20.6NonstationaryModels
20.7CointegratedModels
20.8AsymptoticDistributionofEstimators
andTestCriterion
References
©2001 CRC Press LLC
©2001 CRC Press LLC