Probability for statisticians

著者

• Shorack, Galen R.

書誌事項

Probability for statisticians

Galen R. Shorack

（Springer texts in statistics）

Springer, c2017

2nd ed

注記

Includes bibliographical references (p. 497-500) and index

内容説明・目次

内容説明

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.

目次

PrefaceUse of This TextDefinition of Symbols Chapter 1. Measures Basic Properties of Measures Construction and Extension of Measures Lebesgue Stieltjes Measures Chapter 2. Measurable Functions and Convergence Mappings and -Fields Measurable Functions Convergence Probability, RVs, and Convergence in Law Discussion of Sub -Fields Chapter 3. Integration The Lebesgue Integral Fundamental Properties of Integrals Evaluating and Differentiating Integrals Inequalities Modes of Convergence Chapter 4 Derivatives via Signed Measures Introduction Decomposition of Signed Measures The Radon Nikodym Theorem Lebesgue's Theorem The Fundamental Theorem of Calculus Chapter 5. Measures and Processes on Products Finite-Dimensional Product Spaces Random Vectors on ( , ,P) Countably Infinite Product Probability Spaces Random Elements and Processes on ( , ,P) Chapter 6. Distribution and Quantile Functions Character of Distribution Functions Properties of Distribution Functions The Quantile Transformation Integration by Parts Applied to Moments Important Statistical Quantities Infinite Variances Chapter 7. Independence and Conditional Distributions Independence The Tail -Field Uncorrelated Random Variables Basic Properties of Conditional Expectation Regular Conditional Probability Chapter 8. WLLN, SLLN, LIL, and Series Introduction Borel Cantelli and Kronecker Lemmas Truncation, WLLN, and Review of Inequalities Maximal Inequalities and Symmetrization The Classical Laws of Large Numbers (or, LLNs) Applications of the Laws of Large Numbers Law of the Iterated Logarithm (or, LIL) Strong Markov Property for Sums of IID RVs Convergence of Series of Independent RVs Martinagles Maximal Inequalities, Some with Boundaries Chapter 9. Characteristic Functions and Determining Classes Classical Convergence in Distribution Determining Classes of Functions Characteristic Functions, with Basic Results Uniqueness and Inversion The Continuity Theorem Elementary Complex and Fourier Analysis Esseen's Lemma Distributions on Grids Conditions for O to Be a Characteristic Function Chapter 10. CLTs via Characteristic Functions Introduction Basic Limit Theorems Variations on the Classical CLT Examples of Limiting Distributions Local Limit Theorems Normality Via Winsorization and Truncation Identically Distributed RVs A Converse of the Classical CLT Bootstrapping Bootstrapping with Slowly Winsorization Chapter 11. Infinitely Divisible and Stable Distributions Infinitely Divisible Distributions Stable Distributions Characterizing Stable Laws The Domain of Attraction of a Stable Law Gamma Approximations Edgeworth Expansions Chapter 12. Brownian Motion and Empirical Processes Special Spaces Existence of Processes on (C, C) and (D, D) Brownian Motion and Brownian Bridge Stopping Times Strong Markov Property Embedding a RV in Brownian Motion Barrier Crossing Probabilities Embedding the Partial Sum Process Other Properties of Brownian Motion Various Empirical Processes Inequalities for the Various Empirical Processes Applications Chapter 13. Martingales Basic Technicalities for Martingales Simple Optional Sampling Theorem The Submartingale Convergence Theorem Applications of the S-mg Convergence Theorem Decomposition of a Submartingale Sequence Optional Sampling Applications of Optional Sampling Introduction to Counting Process Martingales CLTs for Dependent RVs Chapter 14. Convergence in Law on Metric Spaces Convergence in Distribution on Metric Spaces Metrics for Convergence in Distribution Chapter 15. Asymptotics Via Empirical Processes Introduction Trimmed and Winsorized Means Linear Rank Statistics and Finite Sampling L-Statistics Appendix A. Special Distributions Elementary Probability Distribution Theory for Statistics Appendix B. General Topology and Hilbert Space General Topology Metric Spaces Hilbert Space Appendix C. More WLLN and CLT Introduction General Moment Estimation Specific Slowly Varying Partial Variance when 2= Specific Tail Relationships Regularly Varying Functions Some Winsorized Variance Comparison Inequalities for Winsorized Quantile Functions ReferencesIndex

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詳細情報

• NII書誌ID(NCID): BB25500183
• ISBN:
  • 9783319522067
• LCCN: 99053670
• 出版国コード: sz
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Cham
• ページ数/冊数: xxii, 510 p.
• 大きさ: 26 cm
• 分類:
  • LCC : QA273
  • DC21 : 519.2
  • DC20 : 519.2
• 件名:
  • LCSH : Probabilities
• 親書誌ID:
  • BA00265563