Description
Mathematical Statistics with Resampling and R
This thoroughly updated third edition combines the latest software applications with the benefits of modern resampling techniques
Resampling helps students understand the meaning of sampling distributions, sampling variability, P-values, hypothesis tests, and confidence intervals. The third edition of Mathematical Statistics with Resampling and R combines modern resampling techniques and mathematical statistics. This book is classroom-tested to ensure an accessible presentation, and uses the powerful and flexible computer language R for data analysis.
This book introduces permutation tests and bootstrap methods to motivate classical inference methods, as well as to be utilized as useful tools in their own right when classical methods are inaccurate or unavailable. The book strikes a balance between simulation, computing, theory, data, and applications.
Throughout the book, new and updated case studies representing a diverse range of subjects, such as flight delays, birth weights of babies, U.S. demographics, views on sociological issues, and problems at Google and Instacart, illustrate the relevance of mathematical statistics to real-world applications.
Changes and additions to the third edition include:
- New and updated case studies that incorporate contemporary subjects like COVID-19
- Several new sections, including introductory material on causal models and regression methods for causal modeling in practice
- Modern terminology distinguishing statistical discernibility and practical importance
- New exercises and examples, data sets, and R code, using dplyr and ggplot2
- A complete instructor’s solutions manual
- A new github site that contains code, data sets, additional topics, and instructor resources
Mathematical Statistics with Resampling and R is an ideal textbook for undergraduate and graduate students in mathematical statistics courses, as well as practitioners and researchers looking to expand their toolkit of resampling and classical techniques.
Table of Contents
Preface xiii
1 Data and Case Studies 1
1.1 Case Study: Flight Delays 1
1.2 Case Study: Birth Weights of Babies 2
1.3 Case Study: Verizon Repair Times 3
1.4 Case Study: Iowa Recidivism 4
1.5 Sampling 5
1.6 Parameters and Statistics 6
1.7 Case Study: General Social Survey 7
1.8 Sample Surveys 8
1.9 Case Study: Beer and Hot Wings 9
1.10 Case Study: Black Spruce Seedlings 10
1.11 Studies 11
1.12 Google Interview Question: Mobile Ads Optimization 13
Exercises 16
2 Exploratory Data Analysis 21
2.1 Basic Plots 21
2.2 Numeric Summaries 25
2.2.1 Center 25
2.2.2 Spread 26
2.2.3 Shape 26
2.3 Boxplots 27
2.4 Quantiles and Normal Quantile Plots 29
2.5 Empirical Cumulative Distribution Functions 34
2.6 Scatter Plots 36
2.7 Skewness and Kurtosis 38
Exercises 39
3 Introduction to Hypothesis Testing: Permutation Tests 45
3.1 Introduction to Hypothesis Testing 45
3.2 Hypotheses 46
3.3 Permutation Tests 50
3.3.1 Implementation Issues 54
3.3.2 One-Sided and Two-Sided Tests 58
3.3.3 Other Statistics 59
3.3.4 Conditions 62
3.3.5 Remark on Terminology 65
3.4 Matched Pairs 66
3.5 Cause and Effect 67
Exercises 71
4 Sampling Distributions 77
4.1 Sampling Distributions 77
4.2 Calculating Sampling Distributions 82
4.3 The Central Limit Theorem 85
4.3.1 CLT for Binomial Data 88
4.3.2 Continuity Correction for Discrete Random Variables 90
4.3.3 Accuracy of the Central Limit Theorem* 92
4.3.4 CLT for Sampling Without Replacement 93
Exercises 93
5 Introduction to Confidence Intervals: The Bootstrap 103
5.1 Introduction to the Bootstrap 103
5.2 The Plug-in Principle 109
5.2.1 Estimating the Population Distribution 110
5.2.2 How Useful Is the Bootstrap Distribution? 112
5.3 Bootstrap Percentile Intervals 115
5.4 Two Sample Bootstrap 116
5.4.1 Matched Pairs 122
5.5 Other Statistics 123
5.6 Bias 126
5.7 Monte Carlo Sampling 130
5.8 Accuracy of Bootstrap Distributions 131
5.8.1 Sample Mean, Large Sample Size 131
5.8.2 Sample Mean: Small Sample Size 132
5.8.3 Sample Median 134
5.8.4 Mean–Variance Relationship 135
5.9 How Many Bootstrap Samples Are Needed? 136
Exercises 137
6 Estimation 147
6.1 Maximum Likelihood Estimation 147
6.1.1 Maximum Likelihood for Discrete Distributions 148
6.1.2 Maximum Likelihood for Continuous Distributions 150
6.1.3 Maximum Likelihood for Multiple Parameters 155
6.2 Method of Moments 158
6.3 Properties of Estimators 160
6.3.1 Unbiasedness 161
6.3.2 Efficiency 164
6.3.3 Mean Square Error 167
6.3.4 Consistency 169
6.3.5 Transformation Invariance* 171
6.3.6 Asymptotic Normality of MLE* 173
6.4 Statistical Practice 174
6.4.1 Are You Asking the Right Question? 175
6.4.2 Weights 175
Exercises 176
7 More Confidence Intervals 183
7.1 Confidence Intervals for Means 183
7.1.1 Confidence Intervals for a Mean, Variance Known 183
7.1.2 Confidence Intervals for a Mean, Variance Unknown 188
7.1.3 Confidence Intervals for a Difference in Means 195
7.1.4 Matched Pairs, Revisited 201
7.2 Confidence Intervals Using Pivots 201
7.2.1 Location and Scale Parameters* 205
7.3 One-Sided Confidence Intervals 209
7.4 Confidence Intervals for Proportions 211
7.4.1 Agresti–Coull Intervals for a Proportion 214
7.4.2 Confidence Interval for a Difference of Proportions 215
7.5 Bootstrap Confidence Intervals 216
7.5.1 T Confidence Intervals Using Bootstrap Standard Errors 216
7.5.2 Bootstrap t Confidence Intervals 217
7.5.3 Comparing Bootstrap t and Formula t Confidence Intervals 223
7.6 Confidence Interval Properties 224
7.6.1 Confidence Interval Accuracy 224
7.6.2 Confidence Interval Length 225
7.6.3 Transformation Invariance 225
7.6.4 Ease of Use and Interpretation 225
7.6.5 Research Needed 226
7.7 The Delta Method* 226
Exercises 230
8 More Hypothesis Testing 245
8.1 Hypothesis Tests for Means and Proportions: One Population 245
8.1.1 A Single Mean 245
8.1.2 One Proportion 248
8.2 Bootstrap t Tests 250
8.3 Hypothesis Tests for Means and Proportions: Two Populations 252
8.3.1 Comparing Two Means 252
8.3.2 Comparing Two Proportions 255
8.3.3 Matched Pairs for Proportions 259
8.4 Type I and Type II Errors 261
8.4.1 Type I Errors 262
8.4.2 Type II Errors and Power 267
8.4.3 P-Values Versus Critical Regions 272
8.4.4 Relationship Between Confidence Intervals and Hypothesis Tests 273
8.5 Interpreting Test Results 276
8.5.1 Terminology 277
8.5.2 Arbitrary Thresholds 277
8.5.3 Statistical Discernibility Versus Practical Importance 277
8.5.4 Negative Results 278
8.5.5 Inflated False Positive Rate 279
8.6 Likelihood Ratio Tests 281
8.6.1 Simple Hypotheses and the Neyman–Pearson Lemma 281
8.6.2 Likelihood Ratio Tests for Composite Hypotheses 285
8.7 Statistical Practice 289
8.7.1 More Campaigns with No Clicks and No Conversions 293
Exercises 294
9 Regression 309
9.1 Covariance 309
9.2 Correlation 313
9.3 Least Squares Regression 316
9.3.1 Regression toward the Mean 320
9.3.2 Variation 321
9.3.3 Diagnostics 323
9.3.4 Multiple Regression 328
9.4 The Simple Linear Model 329
9.4.1 Inference for α and β 333
9.4.2 Inference for the Response 336
9.4.3 Comments About Conditions for the Linear Model 340
9.5 Resampling Correlation and Regression 342
9.5.1 Permutation Tests 345
9.5.2 Bootstrap Case Study: Bushmeat 346
9.6 Logistic Regression 350
9.6.1 Inference for Logistic Regression 355
Exercises 357
10 Categorical Data 367
10.1 Independence in Contingency Tables 367
10.2 Permutation Test of Independence 369
10.3 Chi-Square Test of Independence 371
10.3.1 Model for Chi-Square Test of Independence 373
10.3.2 2 × 2Tables 376
10.3.3 Fisher’s Exact Test 378
10.3.4 Conditioning 379
10.4 Chi-Square Test of Homogeneity 380
10.5 Goodness-of-Fit Tests 382
10.5.1 All Parameters Known 382
10.5.2 Some Parameters Estimated 385
10.6 Chi-Square and the Likelihood Ratio* 388
Exercises 389
11 Bayesian Methods 399
11.1 Bayes Theorem 400
11.2 Binomial Data: Discrete Prior Distributions 400
11.3 Binomial Data: Continuous Prior Distributions 408
11.4 Continuous Data 414
11.5 Sequential Data 417
Exercises 421
12 One-Way ANOVA 429
12.1 Comparing Three or More Populations 429
12.1.1 The ANOVA F Test 429
12.1.2 A Permutation Test Approach 438
Exercises 439
13 Additional Topics 443
13.1 Smoothed Bootstrap 444
13.1.1 Kernel Density Estimate 444
13.2 Parametric Bootstrap 449
13.3 Stratified Sampling 452
13.3.1 Post-stratification 453
13.3.2 Optimal Stratified Sampling 454
13.4 Control Variates and Casual Modeling 455
13.4.1 Control Variates in Experiments 457
13.4.2 Potential Outcomes Framework 460
13.4.3 Observational Data – Causal Modeling 461
13.5 Computational Issues in Bayesian Analysis 462
13.6 Monte Carlo Integration 464
13.7 Importance Sampling 468
13.7.1 Ratio Estimate for Importance Sampling 475
13.7.2 Importance Sampling in Bayesian Applications 478
13.8 The EM Algorithm 483
13.8.1 EM in General 485
Exercises 488
Appendix A Review of Probability 493
A.1 Basic Probability 493
A.2 Mean and Variance 494
A.3 Marginal and Conditional Distributions 496
A.4 The Normal Distribution 497
A.5 The Mean of a Sample of Random Variables 498
A.6 Sums of Normal Random Variables 499
A.7 The Law of Averages 500
A.8 Higher Moments and the Moment Generating Function 501
Appendix B Probability Distributions 505
B. 1 The Bernoulli and Binomial Distributions 505
B. 2 The Multinomial Distribution 506
B. 3 The Geometric Distribution 508
B. 4 The Negative Binomial Distribution 509
B. 5 The Hypergeometric Distribution 510
B. 6 The Poisson Distribution 511
B. 7 The Uniform Distribution 513
B. 8 The Exponential Distribution 513
B. 9 The Gamma Distribution 514
B. 10 The Chi-Square Distribution 517
B. 11 The Student’s t Distribution 520
B. 12 The Beta Distribution 522
B. 13 The F Distribution 523
Exercises 525
Appendix C Distributions Quick Reference 527
Problem Solutions 531
Bibliography 545
Index 553
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