数理统计(第2版)

前言

　　This book is intended for a course entitled Mathematical Statistics offered at the Department of Statistics，University of Wisconsin．Madison．This course，taught in a mathematically rigorous fashion，covers essential materials in statistical theory that a first or second year graduate student typicallY needs to learn as preparation for work on a Ph．D．degree in statistics．The course is designed for two 15-week semesters．with three lecture hours and two discussion hours in each week. Students in this course are assumed to have a good knowledge of advanced calgulus．A course in real analy．sis or measure theory prior to this course is often recommended．Chapter 1 provides a quick overview of important concepts and results in measure-theoretic probability theory that are used as tools in mathematical statistics．Chapter 2 introduces some fundamental concepts in statistics，including statistical models．the principle of SUfIlciency in data reduction，and two statistical approaches adopted throughout the book： statistical decision theory and statistical inference．　　Each of Chapters 3 through 7 provides a detailed study of an important topic in statistical decision theory and inference：Chapter 3 introduces the theory of unbiased estimation；Chapter 4 studies theory and methods in point estimation ander parametric models；Chapter 5 covers point estimation in nonparametric settings；Chapter 6 focuses on hypothesis testing；and Chapter 7 discusses interval estimation and confidence sets．The classical frequentist approach is adopted in this book．although the Bayesian approach is also introduced (§2．3．2，§4．1，§6．4．4，and§7．1．3)．Asymptotic(1arge sample)theory，a crucial part of statistical inference，is studied throughout the book，rather than in a separate chapter．　　About 85％of the book covers classical results in statistical theory that are typically found in textbooks of a similar level．These materials are in the Statistics Department’S Ph．D．qualifying examination syllabus．

书籍目录

Preface to the First Edition

Preface to the Second Edition

Chapter 1.Probability Theory

1.1 Probability Spaces and Random Elements

1.1.1σ－fields and measures

1.1.2 Measurable functions and distributions

1.2 Integration and Differentiation

1.2.1 Integration

1.2.2 Radon.Nikodym derivative

1.3 Distributions and Their Characteristics

1.3.1 Distributions and probability densities

1.3.2 Moments and moment inequalities

1.3.3 Moment generating and characteristic functions

1.4 Conditional Expectations

1.4.1 Conditional expectations

1.4.2 Independence

1.4.3 Conditional distributions

1.4.4 Markov chains and martingales

1.5 Asymptotic Theory

1.5.1 Convergence modes and stochastic orders

1.5.2 Weak convergence

1.5.3 Convergence of transformations

1.5.4 The law of large numbers

1.5.5 The central limit theorem

1.5.6 Edgeworth and Cornish-Fisher expansions

1.6 Exercises

Chapter 2. Fundamentals of Statistics

2.1 Populations，Samples，and Models

2.1.1 Populations and samples

2.1.2 Parametric and nonparametric models

2.1.3 Exponential and location.scale families

2.2 Statistics.Sufficiency,and Completeness

2.2.1 Statistics and their distributions

2.2.2 Sufficiency and minimal sufficiency

2.2.3 Complete statistics

2.3 Statistical Decision Theory

2.3.1 Decision rules，lOSS functions，and risks

2.3.2 Admissibility and optimality

2.4 Statistical Inference

2.4.1 P0il）t estimators

2.4.2 Hypothesis tests

2.4.3 Confidence sets

2.5 Asymptotic Criteria and Inference

2.5.1 Consistency

2.5.2 Asymptotic bias，variance，and mse

2.5.3 Asymptotic inference

2.6 Exercises

Chapter 3.Unbiased Estimation

3.1 The UMVUE

3.1.1 Sufficient and complete statistics

3.1.2 A necessary and.sufficient condition

3.1.3 Information inequality

3.1.4 Asymptotic properties of UMVUE's

3.2 U-Statistics

3.2.1 Some examples

3.2.2 Variances of U-statistics

3.2.3 The projection method

3.3 The LSE in Linear Models

3.3.1 The LSE and estimability

3.3.2 The UMVUE and BLUE

3.3.3 R0bustness of LSE's

3.3.4 Asymptotic properties of LSE's

3.4 Unbiased Estimators in Survey Problems

3.4.1 UMVUE's of population totals

3.4.2 Horvitz-Thompson estimators

3.5 Asymptotically Unbiased Estimators

3.5.1 Functions of unbiased estimators

3.5.2 The method ofmoments

3.5.3 V-statistics

3.5.4 The weighted LSE

3.6 Exercises

Chapter 4.Estimation in Parametric Models

4.1 Bayes Decisions and Estimators

4.1.1 Bayes actions

4.1.2 Empirical and hierarchical Bayes methods

4.1.3 Bayes rules and estimators

4.1.4 Markov chain Mollte Carlo

4.2 Invariance......

4.2.1 One-parameter location families

4.2.2 One-parameter seale families

4.2.3 General location-scale families

4.3 Minimaxity and Admissibility

4.3.1 Estimators with constant risks

4.3.2 Results in one-parameter exponential families

4.3.3 Simultaneous estimation and shrinkage estimators

4.4 The Method of Maximum Likelihood

4.4.1 The likelihood function and MLE's

4.4.2 MLE's in generalized linear models

4.4.3 Quasi-likelihoods and conditional likelihoods

4.5 Asymptotically Efficient Estimation

4.5.1 Asymptotic optimality

4.5.2 Asymptotic efficiency of MLE's and RLE's

4.5.3 Other asymptotically efficient estimators

4.6 Exercises

Chapter 5.Estimation in Nonparametric Models

5.1 Distribution Estimators

5.1.1 Empirical C.d.f.'s in i.i.d.cases

5.1.2 Empirical likelihoods

5.1.3 Density estimation

5.1.4 Semi-parametric methods

5.2 Statistical Functionals

5.2.1 Differentiability and asymptotic normality

5.2.2 L-.M-.and R-estimators and rank statistics

5.3 Linear Functions of Order Statistics

5.3.1 Sample quantiles

5.3.2 R0bustness and efficiency

5.3.3 L-estimators in linear models

5.4 Generalized Estimating Equations

5.4.1 The GEE method and its relationship with others

5.4.2 Consistency of GEE estimators

5.4.3 Asymptotic normality of GEE estimators

5.5 Variance Estimation

5.5.1 The substitution.method

5.5.2 The jackknife

5.5.3 The bootstrap

5.6 Exercises

Chapter 6.Hypothesis Tests

6.1 UMP Tests

6.1.1 The Neyman-Pearson lemma

6.1.2 Monotone likelihood ratio

6.1.3 UMP tests for two-sided hypotheses

6.2 UMP Unbiased Tests

6.2.1 Unbiasedness，similarity，and Neyman structure

6.2.2 UMPU tests in exponential families

6.2.3 UMPU tests in normal families

……

Chapter 7 Confidence Sets

References

List of Notation

List of Abbreviations

Index of Definitions,Main Results,and Examples

Author Index

Subject Index

作者简介

《数理统计(第2版)(英文版)》内容简介：Probability Theory、Probability Spaces and Random Elements、σ－fields and measures、Measurable functions and distributions、Integration and Differentiation、Integration、Radon.Nikodym derivative、Distributions and Their Characteristics、Distributions and probability densities、Moments and moment inequalities、Moment generating and characteristic functions、onditional Expectations、Conditional expectations、Independence、Conditional distributions、Markov chains and martingales、Asymptotic Theory、Convergence modes and stochastic orders等等。

图书封面

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