Jin Zhang编著的《数理统计》内容介绍：This book grew from my lecture notes developed for teaching mathematicalstatistics at Yunnan University (China) and University of Manitoba (Canada). Thecontents and structure of the book are mainly taken from the classical textbookMathematical Statistics: Basic Ideas and Selected Topics (Vol I, 2nd ed. PrenticeHall, 2002) by P. J. Bickel and K. A. Doksum, with reference to other standardtextbooks, such as Mathematical Statistics by K.Knight, Statistical Inference (2nd ed. Duxbury Press, 2002) by G. Casella and R.L. Berger, and Introduction to Mathematical Statistics (6th ed. Prentice Hall, 2005)by R. V. Hogg, J. W. Mckean and A. T. Craig.
The mathematical background necessary for this book is linear algebra andadvance calculus (but no measure theory). It is assumed that the reader is familiarwith basic probability theory and statistical principle.
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 Exponential 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 The 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 Most Powerful Tests
4.4 Confidence Intervals and Regions
4.5 The Duality between Confidence Regions and Hypothesis Tests
4.6 Uniformly Most Accurate Confidence 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
