本书是数理统计方面的经典教材，从数理统计学的初级基本概念及原理开始，详细讲解概率与分布、多元分布、特殊分布、统计推断基础、极大似然法等内容，并且涵盖一些高级主题，如一致性与极限分布、充分性、最优假设检验、正态模型的推断、非参数与稳健统计、贝叶斯统计等.此外，为了帮助读者更好地理解数理统计和巩固所学知识，书中还提供了一些重要的背景材料、大量实例和习题.
本书可以作为高等院校数理统计相关课程的教材，也可供相关专业人员参考使用.

第1章　概率与分布 1
1.1　引论 1
1.2　集合 3
1.2.1　回顾集合论 4
1.2.2　集合函数 7
1.3　概率集函数 12
1.3.1　计数规则 16
1.3.2　概率的附加性质 18
1.4　条件概率与独立性 23
1.4.1　独立性 28
1.4.2　模拟 31
1.5　随机变量 37
1.6　离散随机变量 45
1.6.1　变量变换 47
1.7　连续随机变量 49
1.7.1　分位数 51
1.7.2　变量变换 53
1.7.3　混合离散型和连续型分布 56
1.8　随机变量的期望 60
1.8.1　用R计算期望增益估计 65
1.9　某些特殊期望 68
1.10　重要不等式 78
第2章　多元分布 85
2.1　二元随机变量的分布 85
2.1.1　边际分布 89
2.1.2　期望 93
2.2　二元随机变量变换 100
2.3　条件分布与期望 109
2.4　独立随机变量 117
2.5　相关系数 125
2.6　推广到多个随机变量 134
*2.6.1　多元方差–协方差矩阵 140
2.7　多个随机向量的变换 143
2.8　随机变量的线性组合 151
第3章　某些特殊分布 155
3.1　二项分布及有关分布 155
3.1.1　负二项分布和几何分布 159
3.1.2　多正态分布 160
3.1.3　超几何分布 162
3.2　泊松分布 167
3.3　、2以及分布 173
3.3.1　2分布 178
3.3.2　分布 180
3.4　正态分布 186
*3.4.1　污染正态分布 193
3.5　多元正态分布 198
3.5.1　二元正态分布 198
*3.5.2　多元正态分布的一般情况 199
*3.5.3　应用 206
3.6　t分布与F分布 210
3.6.1　t分布 210
3.6.2　F分布 212
3.6.3　学生定理 214
*3.7　混合分布 218
第4章　基本统计推断 225
4.1　抽样与统计量 225
4.1.1　点估计 226
4.1.2　pmf与pdf的直方图估计 230
4.2　置信区间 238
4.2.1　均值之差的置信区间 241
4.2.2　比例之差的置信区间 243
*4.3　离散分布参数的置信区间 248
4.4　次序统计量 253
4.4.1　分位数 257
4.4.2　分位数置信区间 261
4.5　假设检验介绍 267
4.6　统计检验的深入研究 275
4.6.1　观测的显著性水平：p值 279
4.7　卡方检验 283
4.8　蒙特卡罗方法 292
4.8.1　筛选生成算法 298
4.9　自助法 303
4.9.1　百分位数自助置信区间 303
4.9.2　自助检验法 308
*4.10　分布容许限 315
第5章　一致性与极限分布 321
5.1　依概率收敛 321
5.1.1　抽样和统计量 324
5.2　依分布收敛 327
5.2.1　概率有界 333
5.2.2　Δ方法 334
5.2.3　矩母函数方法 336
5.3　中心极限定理 341
*5.4　推广到多元分布 348
第6章　极大似然法 355
6.1　极大似然估计 355
6.2　拉奥–克拉默下界与有效性 362
6.3　极大似然检验 376
6.4　多参数估计 386
6.5　多参数检验 395
6.6　EM算法 404
第7章　充分性 413
7.1　估计量品质的测量 413
7.2　参数的充分统计量 419
7.3　充分统计量的性质 426
7.4　完备性与唯一性 430
7.5　指数分布类 435
7.6　参数的函数 440
7.6.1　自助标准误差 444
7.7　多参数的情况 447
7.8　最小充分性与从属统计量 454
7.9　充分性、完备性以及独立性 461
第8章　最优假设检验 469
8.1　最大功效检验 469
8.2　一致最大功效检验 479
8.3　似然比检验 487
8.3.1　正态分布均值的似然比检验 488
8.3.2　正态分布方差的似然比检验 495
*8.4　序贯概率比检验 500
*8.5　极小化极大与分类方法 507
8.5.1　极小化极大方法 507
8.5.2　分类 510
第9章　正态线性模型的推断 515
9.1　介绍 515
9.2　单向方差分析 516
9.3　非中心2分布与F分布 522
9.4　多重比较法 525
9.5　双向方差分析 531
9.5.1　因子间的相互作用 534
9.6　回归问题 539
9.6.1　极大似然估计 540
*9.6.2　最小二乘拟合的几何解释 546
9.7　独立性检验 551
9.8　某些二次型的分布 555
9.9　某些二次型的独立性 562
第10章　非参数与稳健统计学 569
10.1　位置模型 569
10.2　样本中位数与符号检验 572
10.2.1　渐近相对有效性 577
10.2.2　基于符号检验的估计方程 582
10.2.3　中位数置信区间 584
10.3　威尔科克森符号秩 586
10.3.1　渐近相对有效性 591
10.3.2　基于威尔科克森符号秩的估计方程 593
10.3.3　中位数置信区间 594
10.3.4　蒙特卡罗调查 595
10.4　曼–惠特尼–威尔科克森方法 598
10.4.1　渐近相对有效性 602
10.4.2　基于MWW的估计方程 604
10.4.3　移位参数Δ的置信区间 604
10.4.4　功效函数的蒙特卡罗调查 605
*10.5　一般秩得分 607
10.5.1　效力 610
10.5.2　基于一般得分的估计方程 612
10.5.3　最优化：最佳估计 612
*10.6　适应方法 619
10.7　简单线性模型 625
10.8　测量关联性 631
10.8.1　肯德尔 631
10.8.2　斯皮尔曼 634
10.9　稳健概念 638
10.9.1　位置模型 638
10.9.2　线性模型 645
第11章　贝叶斯统计 655
11.1　贝叶斯方法 655
11.1.1　先验分布与后验分布 656
11.1.2　贝叶斯点估计 658
11.1.3　贝叶斯区间估计 662
11.1.4　贝叶斯检验方法 663
11.1.5　贝叶斯序贯方法 664
11.2　其他贝叶斯术语及思想 666
11.3　吉布斯抽样器 672
11.4　现代贝叶斯方法 679
11.4.1　经验贝叶斯 682
附录A　数学 687
附录B　R入门 693
附录C　常用分布列表 703
附录D　分布表 707
附录E　参考文献 715
附录F　部分习题答案 721
索引 733



Contents
1 Probability and Distributions ....................................1
1.2 Sets....................................3
1.2.1 Review of SetTheory......................4
1.2.2 Set Functions...........................7
1.3 The Probability SetFunction......................12
1.3.1 Counting Rules..........................16
1.3.2 Additional Properties of Probability..............18
1.4 Conditional Probability and Indepen dence...............23
1.4.1 Independence...........................28
1.4.2 Simulations............................31
1.5 Random Variables............................37
1.6 Discrete Random Variables.......................45
1.6.1 Transformations.........................47
1.7 Continuous Random Variables.....................49
1.7.1 Quantiles.............................51
1.7.2 Transformations.........................53
1.7.3 Mixtures of Discrete and Continuous Type Distributions...56
1.8 Expectation of a Random Variable...................60
1.8.1 R Computation for an Estimation of the Expected Gain...65
1.9 Some Special Expectations.......................68
1.10 Important Inequalities..........................78
2 Multivariate Distributions 85
2.1 Distributions of Two Random Variables................85
2.1.1 Marginal Distributions......................89
2.1.2 Expectation............................93
2.2 Transformations: Bivariate Random Variables.............100
2.3 Conditional Distributions and Expectations..............109
2.4 Independent Random Variables.....................117
2.5 The Correlation Coefficient.......................125
2.6 Extension to Several Random Variables................134
2.6.1 *Multivariate Variance-Covariance Matrix...........140
2.7 Transformations for Several Random Variables............143
2.8 Linear Combinations of Random Variables...............151
3 Some Special Distributions 155
3.1 The Binomial and Related Distributions................155
3.1.1 Negative Binomial and Geometric Distributions........159
3.1.2 Multinomial Distribution....................160
3.1.3 Hypergeometric Distribution..................162
3.2 The Poisson Distribution........................167
3.3 TheΓ,χ2,andβDistributions.....................173
3.3.1 Theχ2-Distribution.......................178
3.3.2 The β-Distribution........................180
3.4 The Normal Distribution.........................186
3.4.1 *Contaminated Normals.....................193
3.5 The Multivariate Normal Distribution.................198
3.5.1 Bivariate Normal Distribution..................198
3.5.2 *Multivariate Normal Distribution, General Case.......199
3.5.3 *Applications...........................206
3.6 t- and F- Distributions..........................210
3.6.1 The t- distribution........................210
3.6.2 The F- distribution........................212
3.6.3 Student’s Theorem........................214
3.7 *Mixture Distributions..........................218
4 Some Elementary Statistical Inferences 225
4.1 Sampling and Statistics.........................225
4.1.1 Point Estimators.........................226
4.1.2 Histogram Estimates of pmfs and pdfs.............230
4.2 ConfidenceIntervals...........................238
4.2.1 Confidence Intervals for Differencein Means..........241
4.2.2 Confidence Interval for Differencein Proportions.......243
4.3 *Confidence Intervals for Parameters of Discrete Distributions....248
4.4 Order Statistics..............................253
4.4.1 Quantiles.............................257
4.4.2 Confidence Intervals for Quantiles...............261
4.5 Introduction to Hypothesis Testing...................267
4.6 Additional Comments About Statistical Tests.............275
4.6.1 Observed Significance Level, p-value..............279
4.7 Chi-Square Tests.............................283
4.8 The Method of Monte Carlo.......................292
4.8.1 Accept–Reject Generation Algorithm..............298
4.9 Bootstrap Procedures..........................303
4.9.1 Percentile Bootstrap Confidence Intervals...........303
4.9.2 Bootstrap Testing Procedures..................308
4.10 *Tolerance Limits for Distributions...................315
5 Consistency and Limiting Distributions 321
5.1 Convergence in Probability.......................321
5.1.1S ampling and Statistics.....................324
5.2 Convergence in Distribution.......................327
5.2.1 Bounded in Probability.....................333
5.2.2 Δ-Method.............................334
5.2.3 Moment Generating Function Technique............336
5.3 Central Limit Theorem.........................341
5.4 *Extensions to Multivariate Distributions...............348
6 Maximum Likelihood Methods 355
6.1 Maximum Likelihood Estimation....................355
6.2 Rao–Cramer Lower Bound and Effciency...............362
6.3 Maximum Likelihood Tests.......................376
6.4 Multiparameter Case: Estimation....................386
6.5 Multiparameter Case: Testing......................395
6.6 The EM Algorithm............................404
7 Sufficiency 413
7.1 Measures of Quality of Estimators...................413
7.2 A Sufficient Statistic for a Parameter..................419
7.3 Properties of a Sufficient Statistic....................426
7.4 Completeness and Uniqueness......................430
7.5 The Exponential Class of Distributions.................435
7.6 Functions of a Parameter........................440
7.6.1 Bootstrap Standard Errors...................444
7.7 The Case of Several Parameters.....................447
7.8 Minimal Sufficiency and Ancillary Statistics..............454
7.9 Sufficiency, Completeness, and Independence.............461
8 Optimal Tests of Hypotheses 469
8.1 Most Powerful Tests...........................469
8.2 Uniformly Most Powerful Tests.....................479
8.3 Likelihood Ratio Tests..........................487
8.3.1 Likelihood Ratio Tests for Testing Means of Normal Distributions..............................488
8.3.2 Likelihood Ratio Tests for Testing Variances of Normal Distributions.............................495
8.4 *The Sequential Probability Ratio Test.................500
8.5 *Minimax and Classification Procedures................507
8.5.1 Minimax Procedures.......................507
8.5.2 Classification...........................510
9 Inferences About Normal Linear Models 515
9.1 Introduction................................515
9.2 One-Way ANOVA............................516
9.3 Noncentralχ2 and F-Distributions...................522
9.4 Multiple Comparisons..........................525
9.5 Two-Way ANOVA............................531
9.5.1 Interaction between Factors...................534
9.6 A Regression Problem..........................539
9.6.1 Maximum Likelihood Estimates.................540
9.6.2 *Geometry of the Least Squares Fit..............546
9.7 A Test of Independence.........................551
9.8 The Distributions of Certain Quadratic Forms.............555
9.9 The Independence of Certain Quadratic Forms............562
10 Nonparametric and Robust Statistics 569
10.1 Location Models.............................569
10.2 Sample Median and the Sign Test....................572
10.2.1 Asymptotic Relative Efficiency.................577
10.2.2 Estimating Equations Basedonthe Sign Test.........582
10.2.3 Confidence Interval for the Median...............584
10.3 Signed-Rank Wilcoxon..........................586
10.3.1 Asymptotic Relative Efficiency.................591
10.3.2 Estimating Equations Basedon Signed-Rank Wilcoxon...593
10.3.3 Confidence Interval for the Median...............594
10.3.4 Monte Carlo Investigation....................595
10.4 Mann–Whitney–Wilcoxon Procedure..................598
10.4.1 Asymptotic Relative Efficiency.................602
10.4.2 Estimating Equations Basedon the Mann–Whitney–Wilcoxon 604
10.4.3 Confidence Interval for the Shift ParameterΔ.........604
10.4.4 Monte Carlo Investigation of Power..............605
10.5 *General RankScores..........................607
10.5.1 Efficacy..............................610
10.5.2 Estimating Equations Based on General Scores........612
10.5.3 Optimizati on: Best Estimates..................612
10.6 *Adaptive Procedures..........................619
10.7 Simple Linear Model...........................625
10.8 Measures of Association.........................631
10.8.1 Kendall’sτ............................631
10.8.2 Spearman’s Rho.........................634
10.9 Robust Concepts.............................638
10.9.1 Location Model..........................638
10.9.2 Linear Model...........................645
11 Bayesian Statistics 655
11.1 Bayesian Procedures...........................655
11.1.1 Prior and Posterior Distributions................656
11.1.2 Bayesian Point Estimation...................658
11.1.3 Bayesian Interval Estimation..................662
11.1.4 Bayesian Testing Procedures..................663
11.1.5 Bayesian Sequential Procedures.................664
11.2 More Bayesian Terminology and Ideas.................666
11.3 Gibbs Sampler..............................672
11.4 Modern Bayesian Methods........................679
11.4.1 Empirical Bayes.........................682
A Mathematical Comments 687
A.1 Regularity Conditions..........................687
A.2 Sequences.................................688
B R Primer 693
B.1 Basics...................................693
B.2 Probability Distributions.........................696
B.3 R Functions................................698
B.4 Loops...................................699
B.5 Inputand Output............................700
B.6 Packages..................................700
C Lists of Common Distributions 703
D Tables of Distributions 707
E References 715
F Answers to Selected Exercises 721
Index 733