A course in large sample theory

Thomas S. Ferguson0412043718, 9780412043710

A Course in Large Sample Theory is presented in four parts. The first treats basic probabilistic notions, the second features the basic statistical tools for expanding the theory, the third contains special topics as applications of the general theory, and the fourth covers more standard statistical topics. Nearly all topics are covered in their multivariate setting.The book is intended as a first year graduate course in large sample theory for statisticians. It has been used by graduate students in statistics, biostatistics, mathematics, and related fields. Throughout the book there are many examples and exercises with solutions. It is an ideal text for self study.

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Table of contents :
Contents……Page 3
Preface……Page 5
Part 1 Basic Probability……Page 8
1 Modes of Convergence……Page 9
2 Partial Converses to Theorem 1……Page 14
3 Convergence in Law……Page 19
4 Laws of Large Numbers……Page 25
5 Central Limit Theorems……Page 32
Part 2 Basic Statistical Large Sample Theory……Page 42
6 Slutsky Theorems……Page 43
7 Functions of the Sample Moments……Page 48
8 The Sample Correlation Coefficient……Page 55
9 Pearson’s Chi-Square……Page 60
10 Asymptotic Power of the Pearson Chi-Square Test……Page 65
Part 3 Special Topics……Page 71
11 Stationary m-Dependent Sequences……Page 72
12 Some Rank Statistics……Page 78
13 Asymptotic Distribution of Sample Quantiles……Page 90
14 Asymptotic Theory of Extreme Order Statistics……Page 97
15 Asymptotic Joint Distributions of Extrema……Page 104
Part 4 Efficient Estimation and Testing……Page 108
16 A Uniform Strong Law of Large Numbers……Page 109
17 Strong Consistency of Maximum-Likelihood Estimates……Page 114
18 Asymptotic Normality of the Maximum-Likelihood Estimate……Page 121
19 The Cramér-Rao Lower Bound……Page 128
20 Asymptotic Efficiency……Page 135
21 Asymptotic Normality of Posterior Distributions……Page 142
22 Asymptotic Distribution of the Likelihood Ratio Test Statistic……Page 146
23 Minimum Chi-Square Estimates……Page 153
24 General Chi-Square Tests……Page 165
Appendix: Solutions to the exercises……Page 174
References……Page 238
Index……Page 240