第1章　概率與分布 1
1.1　引論 1
1.2　集合 3
1.2.1　回顧集合論 4
1.2.2　集合函數(shù) 7
1.3　概率集函數(shù) 12
1.3.1　計(jì)數(shù)規(guī)則 16
1.3.2　概率的附加性質(zhì) 18
1.4　條件概率與獨(dú)立性 23
1.4.1　獨(dú)立性 28
1.4.2　模擬 31
1.5　隨機(jī)變量 37
1.6　離散隨機(jī)變量 45
1.6.1　變量變換 47
1.7　連續(xù)隨機(jī)變量 49
1.7.1　分位數(shù) 51
1.7.2　變量變換 53
1.7.3　混合離散型和連續(xù)型分布 56
1.8　隨機(jī)變量的期望 60
1.8.1　用R計(jì)算期望增益估計(jì) 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　二元隨機(jī)變量的分布 85
2.1.1　邊際分布 89
2.1.2　期望 93
2.2　二元隨機(jī)變量變換 100
2.3　條件分布與期望 109
2.4　獨(dú)立隨機(jī)變量 117
2.5　相關(guān)系數(shù) 125
2.6　推廣到多個(gè)隨機(jī)變量 134
*2.6.1　多元方差–協(xié)方差矩陣 140
2.7　多個(gè)隨機(jī)向量的變換 143
2.8　隨機(jī)變量的線性組合 151
第3章　某些特殊分布 155
3.1　二項(xiàng)分布及有關(guān)分布 155
3.1.1　負(fù)二項(xiàng)分布和幾何分布 159
3.1.2　多正態(tài)分布 160
3.1.3　超幾何分布 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　正態(tài)分布 186
*3.4.1　污染正態(tài)分布 193
3.5　多元正態(tài)分布 198
3.5.1　二元正態(tài)分布 198
*3.5.2　多元正態(tài)分布的一般情況 199
*3.5.3　應(yīng)用 206
3.6　t分布與F分布 210
3.6.1　t分布 210
3.6.2　F分布 212
3.6.3　學(xué)生定理 214
*3.7　混合分布 218
第4章　基本統(tǒng)計(jì)推斷 225
4.1　抽樣與統(tǒng)計(jì)量 225
4.1.1　點(diǎn)估計(jì) 226
4.1.2　pmf與pdf的直方圖估計(jì) 230
4.2　置信區(qū)間 238
4.2.1　均值之差的置信區(qū)間 241
4.2.2　比例之差的置信區(qū)間 243
*4.3　離散分布參數(shù)的置信區(qū)間 248
4.4　次序統(tǒng)計(jì)量 253
4.4.1　分位數(shù) 257
4.4.2　分位數(shù)置信區(qū)間 261
4.5　假設(shè)檢驗(yàn)介紹 267
4.6　統(tǒng)計(jì)檢驗(yàn)的深入研究 275
4.6.1　觀測(cè)的顯著性水平：p值 279
4.7　卡方檢驗(yàn) 283
4.8　蒙特卡羅方法 292
4.8.1　篩選生成算法 298
4.9　自助法 303
4.9.1　百分位數(shù)自助置信區(qū)間 303
4.9.2　自助檢驗(yàn)法 308
*4.10　分布容許限 315
第5章　一致性與極限分布 321
5.1　依概率收斂 321
5.1.1　抽樣和統(tǒng)計(jì)量 324
5.2　依分布收斂 327
5.2.1　概率有界 333
5.2.2　Δ方法 334
5.2.3　矩母函數(shù)方法 336
5.3　中心極限定理 341
*5.4　推廣到多元分布 348
第6章　極大似然法 355
6.1　極大似然估計(jì) 355
6.2　拉奧–克拉默下界與有效性 362
6.3　極大似然檢驗(yàn) 376
6.4　多參數(shù)估計(jì) 386
6.5　多參數(shù)檢驗(yàn) 395
6.6　EM算法 404
第7章　充分性 413
7.1　估計(jì)量品質(zhì)的測(cè)量 413
7.2　參數(shù)的充分統(tǒng)計(jì)量 419
7.3　充分統(tǒng)計(jì)量的性質(zhì) 426
7.4　完備性與唯一性 430
7.5　指數(shù)分布類 435
7.6　參數(shù)的函數(shù) 440
7.6.1　自助標(biāo)準(zhǔn)誤差 444
7.7　多參數(shù)的情況 447
7.8　最小充分性與從屬統(tǒng)計(jì)量 454
7.9　充分性、完備性以及獨(dú)立性 461
第8章　最優(yōu)假設(shè)檢驗(yàn) 469
8.1　最大功效檢驗(yàn) 469
8.2　一致最大功效檢驗(yàn) 479
8.3　似然比檢驗(yàn) 487
8.3.1　正態(tài)分布均值的似然比檢驗(yàn) 488
8.3.2　正態(tài)分布方差的似然比檢驗(yàn) 495
*8.4　序貫概率比檢驗(yàn) 500
*8.5　極小化極大與分類方法 507
8.5.1　極小化極大方法 507
8.5.2　分類 510
第9章　正態(tài)線性模型的推斷 515
9.1　介紹 515
9.2　單向方差分析 516
9.3　非中心2分布與F分布 522
9.4　多重比較法 525
9.5　雙向方差分析 531
9.5.1　因子間的相互作用 534
9.6　回歸問(wèn)題 539
9.6.1　極大似然估計(jì) 540
*9.6.2　最小二乘擬合的幾何解釋 546
9.7　獨(dú)立性檢驗(yàn) 551
9.8　某些二次型的分布 555
9.9　某些二次型的獨(dú)立性 562
第10章　非參數(shù)與穩(wěn)健統(tǒng)計(jì)學(xué) 569
10.1　位置模型 569
10.2　樣本中位數(shù)與符號(hào)檢驗(yàn) 572
10.2.1　漸近相對(duì)有效性 577
10.2.2　基于符號(hào)檢驗(yàn)的估計(jì)方程 582
10.2.3　中位數(shù)置信區(qū)間 584
10.3　威爾科克森符號(hào)秩 586
10.3.1　漸近相對(duì)有效性 591
10.3.2　基于威爾科克森符號(hào)秩的估計(jì)方程 593
10.3.3　中位數(shù)置信區(qū)間 594
10.3.4　蒙特卡羅調(diào)查 595
10.4　曼–惠特尼–威爾科克森方法 598
10.4.1　漸近相對(duì)有效性 602
10.4.2　基于MWW的估計(jì)方程 604
10.4.3　移位參數(shù)Δ的置信區(qū)間 604
10.4.4　功效函數(shù)的蒙特卡羅調(diào)查 605
*10.5　一般秩得分 607
10.5.1　效力 610
10.5.2　基于一般得分的估計(jì)方程 612
10.5.3　最優(yōu)化：最佳估計(jì) 612
*10.6　適應(yīng)方法 619
10.7　簡(jiǎn)單線性模型 625
10.8　測(cè)量關(guān)聯(lián)性 631
10.8.1　肯德?tīng)?631
10.8.2　斯皮爾曼 634
10.9　穩(wěn)健概念 638
10.9.1　位置模型 638
10.9.2　線性模型 645
第11章　貝葉斯統(tǒng)計(jì) 655
11.1　貝葉斯方法 655
11.1.1　先驗(yàn)分布與后驗(yàn)分布 656
11.1.2　貝葉斯點(diǎn)估計(jì) 658
11.1.3　貝葉斯區(qū)間估計(jì) 662
11.1.4　貝葉斯檢驗(yàn)方法 663
11.1.5　貝葉斯序貫方法 664
11.2　其他貝葉斯術(shù)語(yǔ)及思想 666
11.3　吉布斯抽樣器 672
11.4　現(xiàn)代貝葉斯方法 679
11.4.1　經(jīng)驗(yàn)貝葉斯 682
附錄A　數(shù)學(xué) 687
附錄B　R入門(mén) 693
附錄C　常用分布列表 703
附錄D　分布表 707
附錄E　參考文獻(xiàn) 715
附錄F　部分習(xí)題答案 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