Table Of Content

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 authors involved, and it is with great pleasure that we dedicated this volume 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 ©2001 CRC Press LLC ©2001 CRC Press LLC 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 ©2001 CRC Press LLC ©2001 CRC Press LLC 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 ©2001 CRC Press LLC 10BusyPeriodsforSomeQueueswithDeterministic InterarrivalorServiceTimes ClaudeLefèvreandPhilippePicard 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 ©2001 CRC Press LLC ©2001 CRC Press LLC 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 ©2001 CRC Press LLC ©2001 CRC Press LLC 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