Probability for Statisticians

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Springer Science & Business Media, May 2, 2006 - Mathematics - 586 pages
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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.
 

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Contents

Construction and Extension of Measures
12
3
37
4
61
The Fundamental Theorem of Calculus
74
General Topology and Hilbert Space
95
Distribution and Quantile Functions 1 Character of Distribution Functions
107
Properties of Distribution Functions
110
The Quantile Transformation
111
Classical Convergence in Distribution
288
Limit Determining Classes of Functions
292
Brownian Motion and Empirical Processes 1 Special Spaces
295
Existence of Processes on CC and DD
298
Brownian Motion and Brownian Bridge
302
Stopping Times
305
Strong Markov Property
308
Embedding a RV in Brownian Motion
311

Integration by Parts Applied to Moments
115
Important Statistical Quantities
119
Infinite Variance
123
Slowly Varying Partial Variance
127
Specific Tail Relationships
134
Regularly Varying Functions
137
Some Winsorized Variance Comparisons
140
Inequalities for Winsorized Quantile Functions
147
Independence and Conditional Distributions 1 Independence
151
The Tail σField
156
Uncorrelated Random Variables
157
Basic Properties of Conditional Expectation
158
Regular Conditional Probability
168
Conditional Expectations as Projections
174
Special Distributions 1 Elementary Probability
179
Distribution Theory for Statistics
187
Linear Algebra Applications
191
The Multivariate Normal Distribution
199
WLLN SLLN LIL and Series 0 Introduction
203
BorelCantelli and Kronecker Lemmas
204
Truncation WLLN and Review of Inequalities
206
Maximal Inequalities and Symmetrization
210
The Classical Laws of Large Numbers LLNs
215
Applications of the Laws of Large Numbers
223
General Moment Estimation
226
Law of the Iterated Logarithm
235
Strong Markov Property for Sums of IID RVs
239
Convergence of Series of Independent RVs
241
Martingales
246
Maximal Inequalities Some with Boundaries
247
A Uniform SLLN
252
Convergence in Distribution 1 Steins Method for CLTs
255
Winsorization and ˇTruncation
264
Identically Distributed RVs
269
Bootstrapping
274
Bootstrapping with Slowly Increasing Trimming
276
Examples of Limiting Distributions
279
Barrier Crossing Probabilities
314
Embedding the Partial Sum Process
318
Other Properties of Brownian Motion
323
Various Empirical Processes
333
Applications
338
Characteristic Functions 1 Basic Results with Derivation of Common Chfs
341
Uniqueness and Inversion
346
The Continuity Theorem
350
Elementary Complex and Fourier Analysis
352
Esseens Lemma
358
Distributions on Grids
361
Conditions for φ to Be a Characteristic Function
363
CLTs via Characteristic Functions Edgeworth Expansions 390
365
Basic Limit Theorems
366
Variations on the Classical CLT
371
Local Limit Theorems
380
Gamma Approximation
383
Infinitely Divisible and Stable Distributions
399
Stable Distributions
407
Asymptotics via Empirical Proceses
415
Asymptotics via Steins Approach
449
Hoeffdings Combinatorial CLT
458
Martingales
467
The Submartingale Convergence Theorem
473
Applications of the Smg Convergence Theorem
481
Decomposition of a Submartingale Sequence
487
Applications of Optional Sampling
499
DoobMeyer Submartingale Decomposition
511
The Basic Censored Data Martingale
522
CLTs for Dependent RVs
529
Metrics for Convergence in Distribution
540
The Gamma and Digamma Functions
546
Examples of Statistical Models
555
Asymptotics of Maximum Likelihood Estimation
563
Index
575
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