数理统计

1 Statistical Models and Principles

1.1 Statistical Models

1.1.1 Data and Models

1.1.2 Parameters and Statistics

1.2 Bayesian Models

1.3 The Framework of Decision Theory

1.3.1 Components of the Decision Theory

1.3.2 Bayes and Minimax Criteria

1.4 Prediction

1.5 Sufficiency

1.6 Exponential Families

1.6.1 The One-Parameter Case

1.6.2 The Multiparameter Case

1.6.3 Properties of Exponential Families

1.6.4 Conjugate Families of Prior Distributions

1.7 Exercises

2 Methods of Parameter Estimation

2.1 Essentials of Point Estimation

2.1.1 M-Estimation

2.1.2 The Substitution Principle

2.2 Least" Squares and Maximum Likelihood Methods

2.2.1 Least Squares and Weighted Least Squares Estimation

2.2.2 Maximum Likelihood Estimation

2.3 The MLE in Expcnential Families

2.4 Algorithmic Issues for Parameter Estimation

2.4.1 The Bisection Method

2.4.2 The Coordinate Ascent Method

2.4.3 The Newton-Raphson Algorithm

2.4.4 The EM Algorithm

2.5 Exercises

3 Measures of Performance and Optimality

3.1 Bayes Principle

3.2 Minimax Principle

3.3 Unbiased Estimation

3.4 Tl~e Information Inequality

3.4.1 The One-Parameter Case

3.4.2 The Multiparameter Case

3.5 Exercises

4 Hypothesis Tests and Confidence Regions

4.1 The Framework of Hypothesis Testing

4.2 The Neyman-Pearson Test

4.3 Uniformly Mast Powerful Tests

4.4 Ccnfidence Intervals and Regions

4.5 The Duality between Confidence Regions and Hypothesis Tests

4.6 Uniformly Mast Accurate Ccnfidence Bounds

4.7 Bayesian Formulation of Credible Regions

4.8 Prediction Intervals

4.9 Likelihood Ratio Tests

4.9.1 Introduction

4.9.2 One-Sample Problem for a Normal Distribution

4.9.3 Two-Sample Problem with Equal Variance

4.9.4 Two-Sample Problem with Unequal Variances

4.9.5 Likelihood Ratio Tests for Bivariate Normal Distributions

4.10 Exercises

5 Asymptotic Theories

5.1 Introduction

5.2 Consistency

5.2.1 Consistency in Estimation

5.2.2 Consistency of M-Estimates

5.3 Asymptotics Based on the Delta Method

5.3.1 The Delta Method for Approximations of Moments

5.3.2 The Delta Method for Approximations of Distributions

5.4 Asymptotic Theory in One Dimension

5.4.1 Asymptotic Normality of M-Estimates

5.4.2 Asymptotic Normality and Efficiency of MLEs

5.4.3 One-Sided Tests and Confidence Intervals Based on the MLE

5.5 Asymptotic Theory of the Posterior Distribution

5.6 Exercises

6 Asymptotics in the Multiparameter Case

6.1 Asymptotic Normality in k Dimensions

6.1.1 Asymptotic Normality of M-Estimates

6.1.2 Asymptotic Normality and Efficiency of MLEs

6.2 Large-Sample Tests and Confidence Regions

6.2.1 Asymptotic Distribution of the Likelihood-Ratio Test Statistic

6.2.2 Wald's and Rao's Large-Sample Tests and Confidence Regions

6.3 Large-Sample Tests for Categorical Data

6.3.1 Goodness-of-Fit Tests for Multinomial Models

6.3.2 Goodness-of-Fit Tests for Composite Multinomial Models

6.3.3 The X2 Tests for Contingency Tables

6.4 Exercises

Appendix A: Table of Common Distributions

Appendix B: Statistical Tables

Table 1. The Standard Normal Distribution

Table 2. Distribution of t

Table 3. Distribution of X2

Table 4. Distribution of F

References

Index

《数理统计（英文版）（Mathematical Statistics）》是作者根据多年在国内外高校从事教学和科研的经验，专门为英语为非母语的读者精细撰写的英文专著，系统而简明地介绍了《数理统计》的基本方法和理论，并配有适量精选的习题供读者练习。
　　《数理统计（英文版）（Mathematical Statistics）》共分6章，内容包括：统计数据与模型、参数估计方法、优良性准则、假设检验与置信区域、渐近性理论及多参数渐近性理论。