Mathematical Statistics

by Shao, Jun

• ISBN13: 9780387986746
• ISBN10: 038798674X
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Springer Verlag

Summary

Covers topics in statistical theory essential for graduate students preparing for work on a Ph.D. degree in statistics. DLC: Mathematical statistics.

Author Biography

Jun Shao is Professor of Statistics at the University of Wisconsin, Madison.

Table of Contents

Preface	vii
Chapter 1. Probability Theory	1	(60)
1.1 Probability Spaces and Random Elements	1	(8)
1.1.1 Sigma-fields and measures	1	(5)
1.1.2 Measurable functions and distributions	6	(3)
1.2 Integration and Differentiation	9	(8)
1.2.1 Integration	9	(5)
1.2.2 Radon-Nikodym derivative	14	(3)
1.3 Distributions and Their Characteristics	17	(13)
1.3.1 Useful probability densities	17	(8)
1.3.2 Moments and generating functions	25	(5)
1.4 Conditional Expectations	30	(8)
1.4.1 Conditional expectations	30	(4)
1.4.2 Independence	34	(2)
1.4.3 Conditional distributions	36	(2)
1.5 Asymptotic Theorems	38	(11)
1.5.1 Convergence modes and stochastic orders	38	(4)
1.5.2 Convergence of transformations	42	(3)
1.5.3 The law of large numbers	45	(2)
1.5.4 The central limit theorem	47	(2)
1.6 Exercises	49	(12)
Chapter 2. Fundamentals of Statistics	61	(66)
2.1 Populations, Samples, and Models	61	(9)
2.1.1 Populations and samples	61	(3)
2.1.2 Parametric and nonparametric models	64	(2)
2.1.3 Exponential and location-scale families	66	(4)
2.2 Statistics and Sufficiency	70	(13)
2.2.1 Statistics and their distributions	70	(3)
2.2.2 Sufficiency and minimal sufficiency	73	(6)
2.2.3 Complete statistics	79	(4)
2.3 Statistical Decision Theory	83	(9)
2.3.1 Decision rules, loss functions, and risks	83	(3)
2.3.2 Admissibility and optimality	86	(6)
2.4 Statistical Inference	92	(9)
2.4.1 Point estimators	92	(3)
2.4.2 Hypothesis tests	95	(4)
2.4.3 Confidence sets	99	(2)
2.5 Asymptotic Criteria and Inference	101	(11)
2.5.1 Consistency	102	(3)
2.5.2 Asymptotic bias, variance, and mse	105	(4)
2.5.3 Asymptotic inference	109	(3)
2.6 Exercises	112	(15)
Chapter 3. Unbiased Estimation	127	(66)
3.1 The UMVUE	127	(13)
3.1.1 Sufficient and complete statistics	128	(4)
3.1.2 A necessary and sufficient condition	132	(3)
3.1.3 Information inequality	135	(3)
3.1.4 Asymptotic properties of UMVUE's	138	(2)
3.2 U-Statistics	140	(8)
3.2.1 Some examples	140	(2)
3.2.2 Variances of U-statistics	142	(2)
3.2.3 The projection method	144	(4)
3.3 The LSE in Linear Models	148	(13)
3.3.1 The LSE and estimability	148	(4)
3.3.2 The UMVUE and BLUE	152	(3)
3.3.3 Robustness of LSE's	155	(4)
3.3.4 Asymptotic properties of LSE's	159	(2)
3.4 Unbiased Estimators in Survey Problems	161	(9)
3.4.1 UMVUE's of population totals	161	(4)
3.4.2 Horvitz-Thompson estimators	165	(5)
3.5 Asymptotically Unbiased Estimators	170	(12)
3.5.1 Functions of unbiased estimators	170	(3)
3.5.2 The method of moments	173	(3)
3.5.3 V-statistics	176	(3)
3.5.4 The weighted LSE	179	(3)
3.6 Exercises	182	(11)
Chapter 4. Estimation in Parametric Models	193	(84)
4.1 Bayes Decisions and Estimators	193	(20)
4.1.1 Bayes actions	193	(5)
4.1.2 Empirical and hierarchical Bayes methods	198	(3)
4.1.3 Bayes rules and estimators	201	(6)
4.1.4 Markov chain Monte Carlo	207	(6)
4.2 Invariance	213	(10)
4.2.1 One-parameter location families	213	(4)
4.2.2 One-parameter scale families	217	(2)
4.2.3 General location-scale families	219	(4)
4.3 Minimaxity and Admissibility	223	(12)
4.3.1 Estimators with constant risks	223	(4)
4.3.2 Results in one-parameter exponential families	227	(2)
4.3.3 Simultaneous estimation and shrinkage estimators	229	(6)
4.4 The Method of Maximum Likelihood	235	(13)
4.4.1 The likelihood function and MLE's	235	(6)
4.4.2 MLE's in generalized linear models	241	(4)
4.4.3 Quasi-likelihoods and conditional likelihoods	245	(3)
4.5 Asymptotically Efficient Estimation	248	(13)
4.5.1 Asymptotic optimality	248	(4)
4.5.2 Asymptotic efficiency of MLE's and RLE's	252	(5)
4.5.3 Other asymptotically efficient estimators	257	(4)
4.6 Exercises	261	(16)
Chapter 5. Estimation in Nonparametric Models	277	(68)
5.1 Distribution Estimators	277	(14)
5.1.1 Empirical c.d.f.'s in i.i.d. cases	278	(3)
5.1.2 Empirical likelihoods	281	(7)
5.1.3 Density estimation	288	(3)
5.2 Statistical Functionals	291	(13)
5.2.1 Differentiability and asymptotic normality	291	(5)
5.2.2 L-, M-, R-estimators and rank statistics	296	(8)
5.3 Linear Functions of Order Statistics	304	(8)
5.3.1 Sample quantiles	304	(4)
5.3.2 Robustness and efficiency	308	(3)
5.3.3 L-estimators in linear models	311	(1)
5.4 Generalized Estimating Equations	312	(13)
5.4.1 The GEE method and its relationship with others	313	(4)
5.4.2 Consistency of GEE estimators	317	(4)
5.4.3 Asymptotic normality of GEE estimators	321	(4)
5.5 Variance Estimation	325	(12)
5.5.1 The substitution method	326	(3)
5.5.2 The jackknife	329	(5)
5.5.3 The bootstrap	334	(3)
5.6 Exercises	337	(8)
Chapter 6. Hypothesis Tests	345	(76)
6.1 UMP Tests	345	(11)
6.1.1 The Neyman-Pearson lemma	346	(3)
6.1.2 Monotone likelihood ratio	349	(4)
6.1.3 UMP tests for two-sided hypotheses	353	(3)
6.2 UMP Unbiased Tests	356	(13)
6.2.1 Unbiasedness and similarity	356	(2)
6.2.2 UMPU tests in exponential families	358	(4)
6.2.3 UMPU tests in normal families	362	(7)
6.3 UMP Invariant Tests	369	(11)
6.3.1 Invariance and UMPI tests	369	(5)
6.3.2 UMPI tests in normal linear models	374	(6)
6.4 Tests in Parametric Models	380	(14)
6.4.1 Likelihood ratio tests	380	(3)
6.4.2 Asymptotic tests based on likelihoods	383	(4)
6.4.3 X(2)-tests	387	(5)
6.4.4 Bayes tests	392	(2)
6.5 Tests in Nonparametric Models	394	(12)
6.5.1 Sign, permutation, and rank tests	394	(4)
6.5.2 Kolmogorov-Smirnov and Cramer-von Mises tests	398	(3)
6.5.3 Empirical likelihood ratio tests	401	(3)
6.5.4 Asymptotic tests	404	(2)
6.6 Exercises	406	(15)
Chapter 7. Confidence Sets	421	(68)
7.1 Construction of Confidence Sets	421	(13)
7.1.1 Pivotal quantities	421	(6)
7.1.2 Inverting acceptance regions of tests	427	(3)
7.1.3 The Bayesian approach	430	(2)
7.1.4 Prediction sets	432	(2)
7.2 Properties of Confidence Sets	434	(11)
7.2.1 Lengths of confidence intervals	434	(4)
7.2.2 UMA and UMAU confidence sets	438	(3)
7.2.3 Randomized confidence sets	441	(2)
7.2.4 Invariant confidence sets	443	(2)
7.3 Asymptotic Confidence Sets	445	(8)
7.3.1 Asymptotically pivotal quantities	445	(2)
7.3.2 Confidence sets based on likelihoods	447	(4)
7.3.3 Results for quantiles	451	(2)
7.4 Bootstrap Confidence Sets	453	(14)
7.4.1 Construction of bootstrap confidence intervals	453	(4)
7.4.2 Asymptotic correctness and accuracy	457	(6)
7.4.3 High-order accurate bootstrap confidence sets	463	(4)
7.5 Simultaneous Confidence Intervals	467	(8)
7.5.1 Bonferroni's method	468	(1)
7.5.2 Scheffe's method in linear models	469	(2)
7.5.3 Tukey's method in one-way ANOVA models	471	(2)
7.5.4 Confidence bands for c.d.f.'s	473	(2)
7.6 Exercises	475	(14)
Appendix A. Abbreviations	489	(2)
Appendix B. Notation	491	(2)
References	493	(12)
Author Index	505	(4)
Subject Index	509