Introduction to Mathematical Statistics

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 6
Preface......Page 10
1.1 Introduction......Page 12
1.2 Set Theory......Page 14
1.3 The Probability Set Function......Page 21
1.4 Conditional Probability and Independence......Page 32
1.5 Random Variables......Page 43
1.6 Discrete Random Variables......Page 51
1.6.1 Transformations......Page 53
1.7 Continuous RandomVariables......Page 55
1.7.1 Transformations......Page 57
1.8 Expectation of a Random Variable......Page 63
1.9 Some Special Expectations......Page 68
1.10 Important Inequalities......Page 79
2.1 Distributions of Two Random Variables......Page 84
2.1.1 Expectation......Page 90
2.2 Transformations: Bivariate Random Variables......Page 95
2.3 Conditional Distributions and Expectations......Page 105
2.4 The Correlation Coefficient......Page 113
2.5 Independent Random Variables......Page 121
2.6 Extension to Several Random Variables......Page 128
2.6.1 *Multivariate Variance-Covariance Matrix......Page 134
2.7 Transformations for Several Random Variables......Page 137
2.8 Linear Combinations of Random Variables......Page 145
3.1 The Binomial and Related Distributions......Page 150
3.2 The Poisson Distribution......Page 161
3.3 The Γ, χ2, and β Distributions......Page 167
3.4 The Normal Distribution......Page 179
3.4.1 Contaminated Normals......Page 185
3.5 The Multivariate Normal Distribution......Page 189
3.5.1 *Applications......Page 196
3.6.1 The t-distribution......Page 200
3.6.2 The F-distribution......Page 202
3.6.3 Student’s Theorem......Page 204
3.7 Mixture Distributions......Page 208
4.1 Sampling and Statistics......Page 214
4.1.1 HistogramEstimates of pmfs and pdfs......Page 218
4.2 Confidence Intervals......Page 225
4.2.1 Confidence Intervals for Difierence in Means......Page 228
4.2.2 Confidence Interval for Difference in Proportions......Page 230
4.3 Confidence Intervals for Parameters of Discrete Distributions......Page 234
4.4 Order Statistics......Page 238
4.4.1 Quantiles......Page 242
4.4.2 Confidence Intervals for Quantiles......Page 245
4.5 Introduction to Hypothesis Testing......Page 251
4.6 Additional Comments About Statistical Tests......Page 259
4.7 Chi-Square Tests......Page 265
4.8 The Method of Monte Carlo......Page 272
4.8.1 Accept–Reject Generation Algorithm......Page 279
4.9.1 Percentile Bootstrap Confidence Intervals......Page 284
4.9.2 Bootstrap Testing Procedures......Page 287
4.10 *Tolerance Limits for Distributions......Page 295
5.1 Convergence in Probability......Page 300
5.2 Convergence in Distribution......Page 305
5.2.1 Bounded in Probability......Page 311
5.2.2 Δ-Method......Page 312
5.2.3 Moment Generating Function Technique......Page 314
5.3 Central Limit Theorem......Page 318
5.4 *Extensions to Multivariate Distributions......Page 325
6.1 Maximum Likelihood Estimation......Page 332
6.2 Rao–Cramér Lower Bound and Efficiency......Page 338
6.3 Maximum Likelihood Tests......Page 352
6.4 Multiparameter Case: Estimation......Page 361
6.5 Multiparameter Case: Testing......Page 370
6.6 The EM Algorithm......Page 378
7.1 Measures of Quality of Estimators......Page 386
7.2 A Sufficient Statistic for a Parameter......Page 392
7.3 Properties of a Sufficient Statistic......Page 399
7.4 Completeness and Uniqueness......Page 403
7.5 The Exponential Class of Distributions......Page 408
7.6 Functions of a Parameter......Page 413
7.7 The Case of Several Parameters......Page 418
7.8 Minimal Sufficiency and Ancillary Statistics......Page 426
7.9 Sufficiency, Completeness, and Independence......Page 432
8.1 Most Powerful Tests......Page 440
8.2 Uniformly Most Powerful Tests......Page 450
8.3 Likelihood Ratio Tests......Page 458
8.4 The Sequential Probability Ratio Test......Page 470
8.5.1 Minimax Procedures......Page 477
8.5.2 Classification......Page 480
9.1 Quadratic Forms......Page 484
9.2 One-Way ANOVA......Page 489
9.3 Noncentral X[sup(2)] and F-Distributions......Page 495
9.4 Multiple Comparisons......Page 497
9.5 The Analysis of Variance......Page 501
9.6 A Regression Problem......Page 508
9.7 A Test of Independence......Page 517
9.8 The Distributions of Certain Quadratic Forms......Page 520
9.9 The Independence of Certain Quadratic Forms......Page 527
10.1 Location Models......Page 536
10.2 Sample Median and the Sign Test......Page 539
10.2.1 Asymptotic Relative Efficiency......Page 544
10.2.2 Estimating Equations Based on the Sign Test......Page 549
10.2.3 Confidence Interval for the Median......Page 550
10.3 Signed-Rank Wilcoxon......Page 552
10.3.1 Asymptotic Relative Efficiency......Page 557
10.3.3 Confidence Interval for the Median......Page 560
10.4 Mann–Whitney–Wilcoxon Procedure......Page 562
10.4.1 Asymptotic Relative Efficiency......Page 566
10.4.2 Estimating Equations Based on the Mann–Whitney–Wilcoxon......Page 567
10.4.3 Confidence Interval for the Shift Parameter Δ......Page 568
10.5 General Rank Scores......Page 570
10.5.1 Efficacy......Page 573
10.5.2 Estimating Equations Based on General Scores......Page 574
10.5.3 Optimization: Best Estimates......Page 575
10.6 Adaptive Procedures......Page 582
10.7 Simple Linear Model......Page 587
10.8 Measures of Association......Page 592
10.8.1 Kendall’s T......Page 593
10.8.2 Spearman’s Rho......Page 595
10.9 Robust Concepts......Page 599
10.9.1 Location Model......Page 600
10.9.2 Linear Model......Page 606
11.1 Subjective Probability......Page 616
11.2 Bayesian Procedures......Page 619
11.2.1 Prior and Posterior Distributions......Page 620
11.2.2 Bayesian Point Estimation......Page 623
11.2.3 Bayesian Interval Estimation......Page 626
11.2.4 Bayesian Testing Procedures......Page 627
11.2.5 Bayesian Sequential Procedures......Page 628
11.3 More Bayesian Terminology and Ideas......Page 630
11.4 Gibbs Sampler......Page 637
11.5 Modern Bayesian Methods......Page 643
11.5.1 Empirical Bayes......Page 647
A.1 Regularity Conditions......Page 652
A.2 Sequences......Page 653
B: R Functions......Page 656
C: Tables of Distributions......Page 666
D: Lists of Common Distributions......Page 676
E: References......Page 680
F: Answers to Selected Exercises......Page 684
B......Page 694
C......Page 695
D......Page 696
E......Page 697
H......Page 698
J......Page 699
M......Page 700
N......Page 701
P......Page 702
S......Page 703
T......Page 704
Z......Page 705