Probability for Statisticians

by Shorack, Galen R.; Casella, G.; Fienberg, S.; Olkin, I.

• ISBN13: 9780387989532
• ISBN10: 0387989536
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Springer Nature

Summary

Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986-- 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics.

Table of Contents

Preface	vii
Use of This Text	xiii
Definition of Symbols	xviii
Measures
Basic Properties of Measures	1	(11)
Construction and Extension of Measures	12	(6)
Lebesgue-Stieltjes Measures	18	(3)
Measurable Functions and Convergence
Mappings and σ-Fields	21	(3)
Measurable Functions	24	(5)
Convergence	29	(4)
Probability, RVs, and Convergence in Law	33	(2)
Discussion of Sub σ-Fields	35	(2)
Integration
The Lebesgue Integral	37	(3)
Fundamental Properties of Integrals	40	(4)
Evaluating and Differentiating Integrals	44	(2)
Inequalities	46	(5)
Modes of Convergence	51	(10)
Derivatives via Signed Measures
Decomposition of Signed Measures	61	(5)
The Radon-Nikodym Theorem	66	(4)
Lebesgue's Theorem	70	(4)
The Fundamental Theorem of Calculus	74	(5)
Measures and Processes on Products
Finite-Dimensional Product Spaces	79	(5)
Random Vectors on (Ω, A, P)	84	(2)
Countably Infinite Product Probability Spaces	86	(4)
Random Elements and Processes on (Ω, A, P)	90	(5)
General Topology and Hilbert Space
General Topology	95	(6)
Metric Spaces	101	(3)
Hilbert Space	104	(3)
Distribution and Quantile Functions
Character of Distribution Functions	107	(3)
Properties of Distribution Functions	110	(1)
The Quantile Transformation	111	(4)
Integration by Parts Applied to Moments	115	(4)
Important Statistical Quantities	119	(4)
Infinite Variance	123	(5)
Slowly Varying Partial Variance	128	(7)
Specific Tail Relationships	135	(3)
Regularly Varying Functions	138	(4)
Some Winsorized Variance Comparisons	142	(7)
Inequalities for Winsorized Quantile Functions	149	(4)
Independence and Conditional Distributions
Independence	153	(4)
The Tail σ-Field	157	(2)
Uncorrelated Random Variables	159	(1)
Basic Properties of Conditional Expectation	160	(10)
Regular Conditional Probability	170	(6)
Conditional Expectations as Projections	176	(5)
Special Distributions
Elementary Probability	181	(8)
Distribution Theory for Statistics	189	(4)
Linear Algebra Applications	193	(8)
The Multivariate Normal Distribution	201	(4)
WLLN, SLLN, LIL, and Series
Introduction	205	(1)
Borel-Cantelli and Kronecker Lemmas	206	(2)
Truncation, WLLN, and Review of Inequalities	208	(4)
Maximal Inequalities and Symmetrization	212	(5)
The Classical Laws of Large Numbers, LLNs	217	(8)
Applications of the Laws of Large Numbers	225	(3)
General Moment Estimation	228	(9)
Law of the Iterated Logarithm	237	(4)
Strong Markov Property for Sums of IID RVs	241	(2)
Convergence of Series of Independent RVs	243	(5)
Martingales	248	(1)
Maximal Inequalities, Some with → Boundaries	249	(5)
A Uniform SLLN	254	(3)
Convergence in Distribution
Stein's Method for CLTs	257	(9)
Winsorization and Truncation	266	(5)
Identically Distributed RVs	271	(5)
Bootstrapping	276	(2)
Bootstrapping with Slowly Increasing Trimming	278	(3)
Examples of Limiting Distributions	281	(9)
Classical Convergence in Distribution	290	(4)
Limit Determining Classes of Functions	294	(3)
Brownian Motion and Empirical Processes
Special Spaces	297	(3)
Existence of Processes on (C,C) and (D,D)	300	(4)
Brownian Motion and Brownian Bridge	304	(3)
Stopping Times	307	(3)
Strong Markov Property	310	(3)
Embedding a RV in Brownian Motion	313	(3)
Barrier Crossing Probabilities	316	(4)
Embedding the Partial Sum Process	320	(5)
Other Properties of Brownian Motion	325	(2)
Various Empirical Processes	327	(8)
Inequalities for the Various Empirical Processes	335	(5)
Applications	340	(3)
Characteristics Functions
Basic Results, with Derivation of Common Chfs	343	(5)
Uniqueness and Inversion	348	(4)
The Continuity Theorem	352	(2)
Elementary Complex and Fourier Analysis	354	(6)
Esseen's Lemma	360	(3)
Distributions of Grids	363	(2)
Conditions for ø to Be a Characteristic Function	365	(2)
CLTs via Characteristic Functions
Introduction	367	(1)
Basic Limit Theorems	368	(5)
Variations on the Classical CLT	373	(9)
Local Limit Theorems	382	(3)
Gamma Approximation	385	(8)
Edgeworth Expansions	393	(6)
Approximating the Distribution of h(Xn)	399	(2)
Infinitely Divisble and Stable Distributions
Infinitely Divisible Distributions	401	(8)
Stable Distributions	409	(3)
Characterizing Stable Laws	412	(2)
The Domain of Attraction of a Stable Law	414	(3)
Asymptotics via Empirical Proceses
Introduction	417	(1)
Trimmed and Winsorized Means	418	(10)
Linear Rank Statistics and Finite Sampling	428	(6)
The Bootstrap	434	(5)
L-Statistics	439	(12)
Asymptotics via Stein's Approach
U-Statistics	451	(9)
Hoeffding's Combinatorial CLT	460	(9)
Martingales
Basic Technicalities for Martingales	469	(5)
Simple Optional Sampling Theorem	474	(1)
The Submartingale Convergence Theorem	475	(8)
Applications of the S-mg Convergence Theorem	483	(6)
Decomposition of a Submartingale Sequence	489	(5)
Optional Sampling	494	(7)
Applications of Optional Sampling	501	(2)
Introduction to Counting Process Martingales	503	(10)
Doob-Meyer Submartingale Decomposition	513	(5)
Predictable Processes and ∫0 H dM Martingales	518	(6)
The Basic Censored Data Martingale	524	(7)
CLTs for Dependent RVs	531	(2)
Convergence in Law on Metric Spaces
Convergence in Distribution on Metric Spaces	533	(9)
Metrics for Convergence in Distribution	542	(6)
Appendix A. Distribution Summaries
1. The Gamma and Digamma Functions	548	(5)
2. Maximum Likelihood Estimators and Moments	553	(4)
3. Examples of Statistical Models	557	(8)
4. Asymptotics of Maximum Likelihood Estimation	565	(5)
References	570	(5)
Index	575