Approximation Theorems of Mathematical Statistics (Wiley Series in Probability and Statistics)

Approximation Theorems of Mathematical Statistics......Page 5
Contents......Page 13
1.1. Preliminary Notation and Definitions......Page 19
1.2. Modes of Convergence of a Sequence of Random Variables......Page 24
1.3. Relationships Among the Modes of Convergence......Page 27
1.4. Convergence of Moments; Uniform Integrability......Page 31
1.5. Further Discussion of Convergence in Distribution......Page 34
1.6. Operations on Sequences to Produce Specified Convergence Properties......Page 40
1.7. Convergence Properties of Transformed Sequences......Page 42
1.8. Basic Probability Limit Theorems: The WLLN and SLLN......Page 44
1.9. Basic Probability Limit Theorems : The CLT......Page 46
1.10. Basic Probability Limit Theorems : The LIL......Page 53
1.11. Stochastic Process Formulation of the CLT......Page 55
1.12. Taylor’s Theorem; Differentials......Page 61
1.13. Conditions for Determination of a Distribution by Its Moments......Page 63
1.14. Conditions for Existence of Moments of a Distribution......Page 64
1.15. Asymptotic Aspects of Statistical Inference Procedures......Page 65
1.P. Problems......Page 70
2. The Basic Sample Statistics......Page 73
2.1. The Sample Distribution Function......Page 74
2.2. The Sample Moments......Page 84
2.3. The Sample Quantiles......Page 92
2.4. The Order Statistics......Page 105
2.5. Asymptotic Representation Theory for Sample Quantiles, Order Statistics, and Sample Distribution Functions......Page 109
2.6. Confidence Intervals for Quantiles......Page 120
2.7. Asymptotic Multivariate Normality of Cell Frequency Vectors......Page 125
2.8. Stochastic Processes Associated with a Sample......Page 127
2.P. Problems......Page 131
3 Transformations of Given Statistics......Page 135
3.1. Functions of Asymptotically Normal Statistics : Univariate Case......Page 136
3.2. Examples and Applications......Page 138
3.3. Functions of Asymptotically Normal Vectors......Page 140
3.4. Further Examples and Applications......Page 143
3.5. Quadratic Forms in Asymptotically Multivariate Normal Vectors......Page 146
3.6. Functions of Order Statistics......Page 152
3.P. Problems......Page 154
4.1. Asymptotic Optimality in Estimation......Page 156
4.2. Estimation by the Method of Maximum Likelihood......Page 161
4.3. Other Approaches toward Estimation......Page 168
4.4. Hypothesis Testing by Likelihood Methods......Page 169
4.5. Estimation via Product-Multinomial Data......Page 178
4.6. Hypothesis Testing via Product-Multinomial Data......Page 183
4.P. Problems......Page 187
5. U-Statlstics......Page 189
5.1. Basic Description of U-Statistics......Page 190
5.2. The Variance and Other Moments of a U-Statistic......Page 199
5.3. The Projection of a [/-Statistic on the Basic Observations......Page 205
5.4. Almost Sure Behavior of U-Statistics......Page 208
5.5. Asymptotic Distribution Theory of U-Statistics......Page 210
5.6. Probability Inequalities and Deviation Probabilities for U-Statistics......Page 217
5.7. Complements......Page 221
5.P. Problems......Page 225
6. Von Mises Differentiable Statistical Functions......Page 228
6.1. Statistics Considered as Functions of the Sample Distribution Function......Page 229
6.2. Reduction to a Differential Approximation......Page 232
6.3. Methodology for Analysis of the Differential Approximation......Page 239
6.4. Asymptotic Properties of Differentiable Statistical Functions......Page 243
6.5. Examples......Page 249
6.6. Complements......Page 256
6.P. Problems......Page 259
7.1. Basic Formulation and Examples......Page 261
7.2. Asymptotic Properties of M-Estimates......Page 266
7.3. Complements......Page 275
7.P. Problems......Page 278
8.1. Basic Formulation and Examples......Page 280
8.2. Asymptotic Properties of L-Estimates......Page 289
8.P. Problems......Page 308
9.1. Basic Formulation and Examples......Page 310
9.2. Asymptotic Normality of Simple Linear Rank Statistics......Page 313
9.3. Complements......Page 329
9.P. Problems......Page 330
10.1. Approaches toward Comparison of Test Procedures......Page 332
10.2. The Pitman Approach......Page 334
10.3. The Chemoff lndex......Page 343
10.4. Bahadur’s “Stochastic Comparison”......Page 350
10.5. The Hodges-Lehmann Asymptotic Relative Efficiency......Page 359
10.6. Hoeffding’s Investigation (Multinomial Distributions)......Page 360
10.7. The Rubin-Sethuraman “Bayes Risk” Efficiency......Page 365
10.P. Problems......Page 366
Appendix......Page 369
References......Page 371
Author Index......Page 383
Subject Index......Page 387