A Course in Large Sample Theory

Preface vii
Part 1 Basic Probability 1
1 Modes of Convergence 3
2 Partial Converses to Theorem 1 8
3 Convergence in Law 13
4 Laws of Large Numbers 19
5 Central Limit Theorems 26
Part 2 Basic Statistical Large Sample Theory 37
6 Slutsky Theorems 39
7 Functions of the Sample Moments 44
8 The Sample Correlation Coefficient 51
9 Pearson's Chi-Square 56
10 Asymptotic Power of the Pearson Chi-Square Test 61
Part 3 Special Topics 67
11 Stationary m-Dependent Sequences 69
12 Some Rank Statistics 75
13 Asymptotic Distribution of Sample Quantiles 87
14 Asymptotic Theory of Extreme Order Statistics 94
15 Asymptotic Joint Distributions of Extrema 101
Part 4 Efficient Estimation and Testing 105
16 A Uniform Strong Law of Large Numbers 107
17 Strong Consistency of Maximum-Likelihood Estimates 112
18 Asymptotic Normality of the Maximum-Likelihood
Estimate 119
19 The Cram6r-Rao Lower Bound 126
20 Asymptotic Efficiency 133
21 Asymptotic Normality of Posterior Distributions 140
22 Asymptotic Distribution of the Likelihood Ratio
Test Statistic 144
23 Minimum Chi-Square Estimates 151
24 General Chi-Square Tests 163
Appendix: Solutions to the exercises 172
References 236
Index