数理统计学导论

Preface

1 Probability and Distributio

1.1 Introduction

1.2 Set Theory

1.3 The Probability Set Function

1.4 Conditional Probability and Independence

1.5 Random Variables

1.6 Discrete Random Variables

1.6.1 na formatio

1.7 Continuous Random Variables

1.7.1 naDSformatio

1.8 Expectation of a Random Variable

1.9 Some Special Expectatio

1.10 Important Inequalities

2 Multivariate Distributio

2.1 Distributio of Two Random Variables

2.1.1Expectation

2.2 na formatio ：Bivariate Random Variables

2.3 Conditional Distributio and Expectatio

2.4 The Correlation Coefficient

2.5 Independent Random Variables

2.6 Exte ion to Several Random Variables

2.6.1*Multivariate Variance-Covariance Matrix

2.7 na formatio for Several Random Variables

2.8 Linear Combinatio of Random Variables

3 Some Special Distributio

3.1 The Binomial and Related Distributio

3.2 The Poisson Distribution

3.3 The Г，χ2，andβ Distributio

3.4 The Normal Distribution

3.4.1Contaminated Normals

3.5 The Multivariate Normal Distribution

3.5.1*Applicatio

3.6 t-and F-Distributio

3.6.1 The t-distribution

3.6.2 The F-distribution

3.6.3 Student’S Theorem

3.7 Mixture Distributio

4 Some Elementary Statistical Inferences

4.1 Sampling and Statistics

4.1.1 Histogram Estimates of pmfs and pdfs

4.2 Confidence Intervals

4.2.1Confidence Intervals for Difference in Mea

4.2.2Confidence Interval for Difference in Proportio

4.3 Confidence Intervals for Paramete of Discrete Distributio

4.4 CIrder Statistics

4.4.1Quantiles

4.4.2Confidence Intervals for Quantiles

4.5 Introduction to Hypothesis Testing

4.6 Additional Comments About Statistical Tests

4.7 Chi-Square Tests

4.8 The Method of Monte Carlo

4.8.1 Accept-Reject Generation Algorithm

4.9 Bootstrap Procedures

4.9.1 Percentile Bootstrap Confidence Intervals

4.9.2Bootstrap Testing Procedures

4.10 *Tolerance Limits for Distributio

5 Co istency and Limiting Distributio

5.1 Convergence in Probability

5.2 Convergence in Distribution

5.2.1Bounded in Probability

5.2.2 △-Method

5.2.3 Moment Generating Function Technique

5.3 Central Limit Theorem

5.4 *Exte io to Multivariate Distributio

6 Maximum Likelihood Methods

6.1 Maximum Likeli.hood Estimation

6.2 Rao-Cram6r Lower Bound and E伍ciency

6.3 Maximum Likelihood Tests

6.4 Multiparameter Case：Estimation

6.5 Multiparameter Case：Testing

6.6 The EM Algorithm

7 Sufficiency

7.1 Measures of Quality of Estimato

7.2 A Su伍cient Statistic for a Parameter

7.3 Properties of a Sufficient Statistic

7.4 Completeness and Uniqueness

7.5 The Exponential Class of Distributio

7.6 Functio of a Parameter

7.7 The Cuse of Several Paramete

7.8 Minimal Sufficiency and Ancillary Statistics

7.9 Sufficiency,Completeness.and Independence

8 Optimal Tests of Hypotheses

8.1 Most Powerful Tests

8.2 Uniformly Most Powerful Tests

8.3 Likelihood Ratio Tests

8.4 The Sequential Probability Ratio Test

8.5Minimax and Classification Procedures

8.5.1 Minimax Procedures

8.5.2 Classification

9 Inferences About Normal MOdels

9.1 Quadratic Forms

9.2 One-Way ANOVA

9.3 Noncentralχ2and F-Distributio

9.4 Multiple Compariso

9.5 The Analysis of Variance

9.6 A Regression Problem

9.7 A Test of Independence

9.8 The Distributio of Certain Quadratic Forms

9.9 The Independence of Certain Quadratic Forills

10 Nonparametric and Robust Statistics

10.1 Location Models

10.2 Sample Median and the Sign Test

10.2.1 Asymptotic Relative Efficiency

10.2.2 Estimating Equatio Based on the Sign Test

10.2.3 Confidence Interval for the Median

10.3 Signed-Rank Wilcoxon

10.3.1 Asymptotic Relative Emciency

10.3.2 Estimating Equatio Based on Signed-Rank Wilcoxon

10.3.3 Confidence Interval for the Median

10.4 Mann-Whitnev-Wilcoxon Procedure

10.4.1 Asymptotic Relative Efficiency

10.4.2 Estimating Equatio Based on the Mann-Whitney-Wilcoxon

10.4.3 Confidence Interval for the Shift Parameter △

10.5 General Rank Scores

10.5.1 Efficacy

10.5.2 Estimating Equatio Based on General Scores

10.5.3 0ptimization：Best Estimates

10.6 Adaptive Procedures

10.7 Simple Linear Model

10.8 Measures of Association

10.8.1 Kendall’S т

10.8.2 Spearman’S Rho

10.9 Robust Concepts

10.9.1 Location Model

10.9.2 Linear Model

11 Bayesian Statistics

11.1 Subjective Probability

11.2 Bayesian Procedures

11.2.1 Prior and Posterior Distributio

11.2.2 Bayesian Point Estimation

11.2.3 Bayesian Interval Estimation

11.2.4 Bayesian Testing Procedures

11.2.5 Bayesian Sequential Procedures

11.3 More Bayesian Terminology and Ideas

11.4 Gibbs Sampler

11.5 Modern Bayesian Methods

11.5.1 Empirical Bayes

A Mathematical Comments

A.1 Regularity Conditio

A.2 Sequences

B R Functio

C Tables of Distributio

D Lists of Common Distributio

E References

F A we to Selected Exercisds

Index

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