Shorack Probability for Statisticians

2. Auflage 2017
ISBN: 978-3-319-52207-4
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 510 Seiten, eBook

Reihe: Springer Texts in Statistics

The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.

Zielgruppe

Graduate

Autoren/Hrsg.

Shorack, Galen R.

Inhaltsverzeichnis

PrefaceUse of This TextDefinition of SymbolsChapter 1. MeasuresBasic Properties of MeasuresConstruction and Extension of MeasuresLebesgue Stieltjes MeasuresChapter 2. Measurable Functions and ConvergenceMappings and s-FieldsMeasurable FunctionsConvergenceProbability, RVs, and Convergence in LawDiscussion of Sub s-FieldsChapter 3. IntegrationThe Lebesgue IntegralFundamental Properties of IntegralsEvaluating and Differentiating IntegralsInequalitiesModes of ConvergenceChapter 4 Derivatives via Signed MeasuresIntroductionDecomposition of Signed MeasuresThe Radon Nikodym TheoremLebesgue's TheoremThe Fundamental Theorem of CalculusChapter 5. Measures and Processes on ProductsFinite-Dimensional Product SpacesRandom Vectors on (O,?,P)Countably Infinite Product Probability SpacesRandom Elements and Processes on (O,?,P)Chapter 6. Distribution and Quantile FunctionsCharacter of Distribution FunctionsProperties of Distribution FunctionsThe Quantile TransformationIntegration by Parts Applied to MomentsImportant Statistical QuantitiesInfinite VariancesChapter 7. Independence and Conditional DistributionsIndependenceThe Tail s-FieldUncorrelated Random VariablesBasic Properties of Conditional ExpectationRegular Conditional ProbabilityChapter 8. WLLN, SLLN, LIL, and SeriesIntroductionBorel Cantelli and Kronecker LemmasTruncation, WLLN, and Review of InequalitiesMaximal Inequalities and SymmetrizationThe Classical Laws of Large Numbers (or, LLNs)Applications of the Laws of Large NumbersLaw of the Iterated Logarithm (or, LIL)Strong Markov Property for Sums of IID RVsConvergence of Series of Independent RVsMartinaglesMaximal Inequalities, Some with ? BoundariesChapter 9. Characteristic Functions and Determining ClassesClassical Convergence in DistributionDetermining Classes of FunctionsCharacteristic Functions, with Basic ResultsUniqueness and InversionThe Continuity TheoremElementary Complex and Fourier AnalysisEsseen's LemmaDistributions on GridsConditions for Ø to Be a Characteristic FunctionChapter 10. CLTs via Characteristic FunctionsIntroductionBasic Limit TheoremsVariations on the Classical CLTExamples of Limiting DistributionsLocal Limit TheoremsNormality Via Winsorization and TruncationIdentically Distributed RVsA Converse of the Classical CLTBootstrappingBootstrapping with Slowly ? WinsorizationChapter 11. Infinitely Divisible and Stable DistributionsInfinitely Divisible DistributionsStable DistributionsCharacterizing Stable LawsThe Domain of Attraction of a Stable LawGamma ApproximationsEdgeworth ExpansionsChapter 12. Brownian Motion and Empirical ProcessesSpecial Spaces Existence of Processes on (C, C) and (D, D)Brownian Motion and Brownian BridgeStopping TimesStrong Markov PropertyEmbedding a RV in Brownian MotionBarrier Crossing ProbabilitiesEmbedding the Partial Sum ProcessOther Properties of Brownian MotionVarious Empirical ProcessesInequalities for the Various Empirical ProcessesApplicationsChapter 13. MartingalesBasic Technicalities for MartingalesSimple Optional Sampling TheoremThe Submartingale Convergence TheoremApplications of the S-mg Convergence TheoremDecomposition of a Submartingale SequenceOptional SamplingApplications of Optional SamplingIntroduction to Counting Process MartingalesCLTs for Dependent RVsChapter 14. Convergence in Law on Metric SpacesConvergence in Distribution on Metric SpacesMetrics for Convergence in Distribution Chapter 15. Asymptotics Via Empirical ProcessesIntroductionTrimmed and Winsorized MeansLinear Rank Statistics and Finite SamplingL-StatisticsAppendix A. Special DistributionsElementary ProbabilityDistribution Theory for StatisticsAppendix B. General Topology and Hilbert SpaceGeneral TopologyMetric SpacesHilbert SpaceAppendix C. More WLLN and CLTIntroductionGeneral Moment Estimation SpecificSlowly Varying Partial Variance when s2=8Specific Tail RelationshipsRegularly Varying FunctionsSome Winsorized Variance ComparisonInequalities for Winsorized Quantile FunctionsReferencesIndex

Über Autor(innen)

Galen Shorack, PhD, is Professor Emeritus in the Department of Statistics (of which he was a founding member) and Adjunct Professor in the Department of Mathematics at the University of Washington, Seattle.  He received his Bachelor of Science and Master of Science degrees in Mathematics from the University of Oregon and his PhD in Statistics from Stanford University.  Dr. Shorack's research interests include limit theorems in statistics, the theory of empirical processes, trimming-Winsorizing, and regular variation.  He has served as Associate Editor of the Annals of Mathematical Statistics (Annals of Statistics) and is Fellow of the Institute of Mathematical Statistics.