Serfling R.J. Approximation Theorems of Mathematical Statistics

Wiley, 1980. — 392 p.This paperback reprint of one of the best in the field 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.Preliminary Notation and Definitions
Modes of Convergence of a Sequence of Random Variables
Relationships Among the Modes of Convergence
Convergence of Moments; Uniform Integrability
Further Discussion of Convergence in Distribution
Operations on Sequences to Produce Specified Convergence Properties
Convergence Properties of Transformed Sequences
Basic Probability Limit Theorems: The WLLN and SLLN
Basic Probability Limit Theorems : The CLT
Basic Probability Limit Theorems : The LIL
Stochastic Process Formulation of the CLT
Taylor’s Theorem; Differentials
Conditions for Determination of a Distribution by Its Moments
Conditions for Existence of Moments of a Distribution
Asymptotic Aspects of Statistical Inference Procedures
ProblemsThe Basic Sample Statistics
The Sample Distribution Function
The Sample Moments
The Sample Quantiles
The Order Statistics
Asymptotic Representation Theory for Sample Quantiles, Order Statistics, and Sample Distribution Functions
Confidence Intervals for Quantiles
Asymptotic Multivariate Normality of Cell Frequency Vectors
Stochastic Processes Associated with a Sample
ProblemsTransformations of Given Statistics
Functions of Asymptotically Normal Statistics : Univariate Case
Examples and Applications
Functions of Asymptotically Normal Vectors
Further Examples and Applications
Quadratic Forms in Asymptotically Multivariate Normal Vectors
Functions of Order Statistics
P Problems
Asymptotic Optimality in Estimation
Estimation by the Method of Maximum Likelihood
Other Approaches toward Estimation
Hypothesis Testing by Likelihood Methods
Estimation via Product-Multinomial Data
Hypothesis Testing via Product-Multinomial Data
ProblemsU-Statlstics
Basic Description of U-Statistics
The Variance and Other Moments of a U-Statistic
The Projection of a [/-Statistic on the Basic Observations
Almost Sure Behavior of U-Statistics
Asymptotic Distribution Theory of U-Statistics
Probability Inequalities and Deviation Probabilities for U-Statistics
Complements
ProblemsVon Mises Differentiable Statistical Functions
Statistics Considered as Functions of the Sample Distribution Function
Reduction to a Differential Approximation
Methodology for Analysis of the Differential Approximation
Asymptotic Properties of Differentiable Statistical Functions
Examples
Complements
ProblemsBasic Formulation and Examples
Asymptotic Properties of M-Estimates
Complements
ProblemsBasic Formulation and Examples
Asymptotic Properties of L-Estimates
ProblemsBasic Formulation and Examples
Asymptotic Normality of Simple Linear Rank Statistics
Complements
ProblemsApproaches toward Comparison of Test Procedures
The Pitman Approach
The Chemoff lndex
Bahadur’s Stochastic Comparison
The Hodges-Lehmann Asymptotic Relative Efficiency
Hoeffding’s Investigation (Multinomial Distributions)
The Rubin-Sethuraman Bayes Risk Efficiency
ProblemsAppendixAuthor Index
Subject Index