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Approximation Theorems of Mathematical Statistics
This convenient paperback edition makes a seminal text in statistics accessible to a new generation of students and practitioners. Approximation Theorems of Mathematical Statistics covers a broad range of limit theorems useful in mathematical statistics, along with methods of proof and techniques of application. The manipulation of "probability" theorems to obtain "statistical" theorems is emphasized. Besides a knowledge of these basic statistical theorems, this lucid introduction to the subject imparts an appreciation of the instrumental role of probability theory.
The book makes accessible to students and practicing professionals in statistics, general mathematics, operations research, and engineering the essentials of:
* The tools and foundations that are basic to asymptotic theory in statistics
* The asymptotics of statistics computed from a sample, including transformations of vectors of more basic statistics, with emphasis on asymptotic distribution theory and strong convergence
* Important special classes of statistics, such as maximum likelihood estimates and other asymptotic efficient procedures; W. Hoeffding's U-statistics and R. von Mises's "differentiable statistical functions"
* Statistics obtained as solutions of equations ("M-estimates"), linear functions of order statistics ("L-statistics"), and rank statistics ("R-statistics")
* Use of influence curves
* Approaches toward asymptotic relative efficiency of statistical test procedures
Contents
1 Preliminary Tools and Foundations 1
1.1 Preliminary Notation and Definitions 1
1.2 Modes of Convergence of a Sequence of Random Variables 6
1.3 Relationships Among the Modes of Convergence 9
1.4 Convergence of Moments; Uniform Integrability 13
1.5 Further Discussion of Convergence in Distribution 16
1.6 Operations on Sequences to Produce Specified Convergence Properties 22
1.7 Convergence Properties of Transformed Sequences 24
1.8 Basic Probability Limit Theorems: The WLLN and SLLN 26
1.9 Basic Probability Limit Theorems: The CLT 28
1.10 Basic Probability Limit Theorems: The LIL 35
1.11 Stochastic Process Formulation of the CLT 37
1.12 Taylor's Theorem; Differentials 43
1.13 Conditions for Determination of a Distribution by Its Moments 45
1.14 Conditions for Existence of Moments of a Distribution 46
1.15 Asymptotic Aspects of Statistical Inference Procedures 47
1.P Problems 52
2 The Basic Sample Statistics 55
2.1 The Sample Distribution Function 56
2.2 The Sample Moments 66
2.3 The Sample Quantiles 74
2.4 The Order Statistics 87
2.5 Asymptotic Representation Theory for Sample Quantiles Order Statistics and Sample Distribution Functions 91
2.6 Confidence Intervals for Quantiles 102
2.7 Asymptotic Multivariate Normality of Cell Frequency Vectors 107
2.8 Stochastic Processes Associated with a Sample 109
2.P Problems 113
3 Transformations of Given Statistics 117
3.1 Functions of Asymptotically Normal Statistics: Univariate Case 118
3.2 Examples and Applications 120
3.3 Functions of Asymptotically Normal Vectors 122
3.4 Further Examples and Applications 125
3.5 Quadratic Forms in Asymptotically Multivariate Normal Vectors 128
3.6 Functions of Order Statistics 134
3.P Problems 136
4 Asymptotic Theory in Parametric Inference 138
4.1 Asymptotic Optimality in Estimation 138
4.2 Estimation by the Method of Maximum Likelihood 143
4.3 Other Approaches toward Estimation 150
4.4 Hypothesis Testing by Likelihood Methods 151
4.5 Estimation via Product-Multinomial Data 160
4.6 Hypothesis Testing via Product-Multinomial Data 165
4.P Problems 169
5 U-Statistics 171
5.1 Basic Description of U-Statistics 172
5.2 The Variance and Other Moments of a U-Statistic 181
5.3 The Projection of a U-Statistic on the Basie Observations 187
5.4 Almost Sure Behavior of U-Statistics 190
5.5 Asymptotic Distribution Theory of U-Statistics 192
5.6 Probability Inequalities and Deviation Probabilities for U-Statistics 199
5.7 Complements 203
5.P Problems 207
6 Von Mises Differentiable Statistical Functions 210
6.1 Statistics Considered as Functions of the Sample Distribution Function 211
6.2 Reduction to a Differential Approximation 214
6.3 Methodology for Analysis of the Differential Approximation 221
6.4 Asymptotic Properties of Differentiable Statistical Functions 225
6.5 Examples 231
6.6 Complements 238
6.P Problems 241
7 M-Estimates 243
7.1 Basic Formulation and Examples 243
7.2 Asymptotic Properties of M-Estimates 248
7.3 Complements 257
7.P Problems 260
8 L-Estimates
8.1 Basic Formulation and Examples 262
8.2 Asymptotic Properties of L-Estimates 271
8.P Problems 290
9 R-Estimates
9.1 Basic Formulation and Examples 292
9.2 Asymptotic Normality of Simple Linear Rank Statistics 295
9.3 Complements 311
9.P Problems 312
10 Asymptotic Relative Efficiency
10.1 Approaches toward Comparison of Test Procedures 314
10.2 The Pitman Approach 316
10.3 The Chernoff Index 325
10.4 Bahadur's "Stochastic Comparison," 332
10.5 The Hodges-Lehmann Asymptotic Relative Efficiency 341
10.6 Hoeffding's Investigation (Multinomial Distributions) 342
10.7 The Rubin‒Sethuraman "Bayes Risk" Efficiency 347
I0.P Problems 348
Appendix 351
References 553
Author Index 365
Subject Index 369
