Mathematical statistics and data analysis

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 5
Preface......Page 12
1.1 Introduction......Page 18
1.2 Sample Spaces......Page 19
1.3 Probability Measures......Page 21
1.4 Computing Probabilities: Counting Methods......Page 23
1.4.1 The Multiplication Principle......Page 24
1.4.2 Permutations and Combinations......Page 26
1.5 Conditional Probability......Page 33
1.6 Independence......Page 40
1.8 Problems......Page 43
2.1 Discrete Random Variables......Page 52
2.1.1 Bernoulli Random Variables......Page 54
2.1.2 The Binomial Distribution......Page 55
2.1.3 The Geometric and Negative Binomial Distributions......Page 57
2.1.5 The Poisson Distribution......Page 59
2.2 Continuous Random Variables......Page 64
2.2.1 The Exponential Density......Page 67
2.2.2 The Gamma Density......Page 70
2.2.3 The Normal Distribution......Page 71
2.3 Functions of a Random Variable......Page 75
2.5 Problems......Page 81
3.1 Introduction......Page 88
3.2 Discrete Random Variables......Page 89
3.3 Continuous Random Variables......Page 92
3.4 Independent Random Variables......Page 101
3.5.1 The Discrete Case......Page 104
3.5.2 The Continuous Case......Page 105
3.6.1 Sums and Quotients......Page 113
3.6.2 The General Case......Page 116
3.7 Extrema and Order Statistics......Page 121
3.8 Problems......Page 124
4.1 The Expected Value of a Random Variable......Page 133
4.1.1 Expectations of Functions of Random Variables......Page 138
4.1.2 Expectations of Linear Combinations of Random Variables......Page 141
4.2 Variance and Standard Deviation......Page 147
4.2.1 A Model for Measurement Error......Page 152
4.3 Covariance and Correlation......Page 155
4.4.1 Definitions and Examples......Page 164
4.4.2 Prediction......Page 169
4.5 The Moment-Generating Function......Page 172
4.6 Approximate Methods......Page 178
4.7 Problems......Page 183
5.2 The Law of Large Numbers......Page 194
5.3 Convergence in Distribution and the Central Limit Theorem......Page 198
5.4 Problems......Page 205
6.1 Introduction......Page 209
6.3 The Sample Mean and the Sample Variance......Page 212
6.4 Problems......Page 215
7.1 Introduction......Page 216
7.2 Population Parameters......Page 217
7.3 Simple Random Sampling......Page 219
7.3.1 The Expectation and Variance of the Sample Mean......Page 220
7.3.2 Estimation of the Population Variance......Page 227
7.3.3 The Normal Approximation to the Sampling Distribution of X......Page 231
7.4 Estimation of a Ratio......Page 237
7.5.1 Introduction and Notation......Page 244
7.5.2 Properties of Stratified Estimates......Page 245
7.5.3 Methods of Allocation......Page 249
7.6 Concluding Remarks......Page 255
7.7 Problems......Page 256
8.2 Fitting the Poisson Distribution to Emissions of Alpha Particles......Page 272
8.3 Parameter Estimation......Page 274
8.4 The Method of Moments......Page 277
8.5 The Method of Maximum Likelihood......Page 284
8.5.1 Maximum Likelihood Estimates of Multinomial Cell Probabilities......Page 289
8.5.2 Large Sample Theory for Maximum Likelihood Estimates......Page 291
8.5.3 Con.dence Intervals from Maximum Likelihood Estimates......Page 296
8.6 The Bayesian Approach to Parameter Estimation......Page 302
8.6.1 Further Remarks on Priors......Page 311
8.6.2 Large Sample Normal Approximation to the Posterior......Page 313
8.6.3 Computational Aspects......Page 314
8.7 Efficiency and the Cramér-Rao Lower Bound......Page 315
8.7.1 An Example: The Negative Binomial Distribution......Page 319
8.8 Sufficieny......Page 322
8.8.1 A Factorization Theorem......Page 323
8.8.2 The Rao-Blackwell Theorem......Page 327
8.9 Concluding Remarks......Page 328
8.10 Problems......Page 329
9.1 Introduction......Page 346
9.2 The Neyman-Pearson Paradigm......Page 348
9.2.1 Specification of the Significance Level and the Concept of a p-value......Page 351
9.2.2 The Null Hypothesis......Page 352
9.2.3 Uniformly Most Powerful Tests......Page 353
9.3 The Duality of Confidence Intervals and Hypothesis Tests......Page 354
9.4 Generalized Likelihood Ratio Tests......Page 356
9.5 Likelihood Ratio Tests for the Multinomial Distribution......Page 358
9.6 The Poisson Dispersion Test......Page 364
9.7 Hanging Rootograms......Page 366
9.8 Probability Plots......Page 369
9.9 Tests for Normality......Page 375
9.10 Concluding Remarks......Page 378
9.11 Problems......Page 379
10.1 Introduction......Page 394
10.2.1 The Empirical Cumulative Distribution Function......Page 395
10.2.2 The Survival Function......Page 397
10.2.3 Quantile-Quantile Plots......Page 402
10.3 Histograms, Density Curves, and Stem-and-Leaf Plots......Page 406
10.4 Measures of Location......Page 409
10.4.1 The Arithmetic Mean......Page 410
10.4.2 The Median......Page 412
10.4.4 M Estimates......Page 414
10.4.5 Comparison of Location Estimates......Page 415
10.4.6 Estimating Variability of Location Estimates by the Bootstrap......Page 416
10.5 Measures of Dispersion......Page 418
10.6 Boxplots......Page 419
10.7 Exploring Relationships with Scatterplots......Page 421
10.8 Concluding Remarks......Page 424
10.9 Problems......Page 425
11.1 Introduction......Page 437
11.2.1 Methods Based on the Normal Distribution......Page 438
11.2.2 Power......Page 450
11.2.3 A Nonparametric Method—The Mann-Whitney Test......Page 452
11.2.4 Bayesian Approach......Page 460
11.3 Comparing Paired Samples......Page 461
11.3.1 Methods Based on the Normal Distribution......Page 463
11.3.2 A Nonparametric Method—The Signed Rank Test......Page 465
11.3.3 An Example—Measuring Mercury Levels in Fish......Page 467
11.4.1 Mammary Artery Ligation......Page 469
11.4.3 The Lanarkshire Milk Experiment......Page 470
11.4.4 The Portacaval Shunt......Page 471
11.4.5 FD&C Red No. 40......Page 472
11.4.6 Further Remarks on Randomization......Page 473
11.4.7 Observational Studies, Confounding, and Bias in Graduate Admissions......Page 474
11.4.8 Fishing Expeditions......Page 475
11.6 Problems......Page 476
12.2 The One-Way Layout......Page 494
12.2.1 Normal Theory; the F Test......Page 495
12.2.2 The Problem of Multiple Comparisons......Page 502
12.2.3 A Nonparametric Method—The Kruskal-Wallis Test......Page 505
12.3.1 Additive Parametrization......Page 506
12.3.2 Normal Theory for the Two-Way Layout......Page 509
12.3.3 Randomized Block Designs......Page 517
12.3.4 A Nonparametric Method—Friedman\'s Test......Page 520
12.4 Concluding Remarks......Page 521
12.5 Problems......Page 522
13.2 Fisher\'s Exact Test......Page 531
13.3 The Chi-Square Test of Homogeneity......Page 533
13.4 The Chi-Square Test of Independence......Page 537
13.5 Matched-Pairs Designs......Page 540
13.6 Odds Ratios......Page 543
13.8 Problems......Page 547
14.1 Introduction......Page 559
14.2.1 Statistical Properties of the Estimated Slope and Intercept......Page 564
14.2.2 Assessing the Fit......Page 567
14.2.3 Correlation and Regression......Page 577
14.3 The Matrix Approach to Linear Least Squares......Page 581
14.4.1 Vector-Valued Random Variables......Page 584
14.4.2 Mean and Covariance of Least Squares Estimates......Page 590
14.4.5 Inference about β......Page 593
14.5 Multiple Linear Regression—An Example......Page 597
14.6 Conditional Inference, Unconditional Inference, and the Bootstrap......Page 602
14.7 Local Linear Smoothing......Page 604
14.9 Problems......Page 608
APPENDIX A: Common Distributions......Page 622
APPENDIX B: Tables......Page 625
Bibliography......Page 646
Answers to Selected Problems......Page 653
Author Index......Page 669
Applications Index......Page 672
Subject Index......Page 675
Credits......Page 685