Mathematical Statistics with Applications in R, Fourth Edition offers a modern calculus-based theoretical introduction to mathematical statistics and applications that spans numerous foundational and essential concepts in the field. The book covers many modern statistical computational and simulation concepts, including Exploratory Data Analysis, the Jackknife, bootstrap methods, the EM algorithms, and Markov chain Monte Carlo (MCMC) methods such as the Metropolis algorithm, Metropolis-Hastings algorithm and the Gibbs sampler. The final chapter of the book provides a step-by-step approach to modelling, analysis, and interpretation data from real-world applications, from the environment and cyber security to health and finance.
By combining discussion on the theory of statistics with a wealth of engaging, real-world applications, this book helps students approach statistical problem-solving in a logical manner with accessible, step-by-step procedures on relatable topics. Computational aspects are covered through R and SAS examples.
1. 2.1 INTRODUCTION - p. 47 2. 2.2 RANDOM EVENTS AND PROBABILITY - p. 48 3. EXERCISES 2.2 - p. 52 4. 2.3 COUNTING TECHNIQUES AND CALCULATION OF PROBABILITIES - p. 55 5. EXERCISES 2.3 - p. 60 6. 2.4 CONDITIONAL PROBABILITY, INDEPENDENCE, AND BAYES’ RULE - p. 61 7. EXERCISES 2.4 - p. 67 8. 2.5 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS - p. 71 9. EXERCISES 2.5 - p. 77 10. 2.6 MOMENTS AND MOMENT-GENERATING FUNCTIONS - p. 79 11. EXERCISES 2.6 - p. 90 12. 2.7 CHAPTER SUMMARY - p. 92 13. 2.8 COMPUTER EXAMPLES (OPTIONAL) - p. 93 14. PROJECTS FOR CHAPTER 2 - p. 96 15. Chapter 3. Probability Distributions - p. 99 16. Chapter outline - p. 99 17. 3.1 INTRODUCTION - p. 100 18. 3.2 SPECIAL DISTRIBUTION FUNCTIONS - p. 100 19. EXERCISES 3.2 - p. 117 20. 3.3 JOINT PROBABILITY DISTRIBUTIONS - p. 121 21. EXERCISES 3.3 - p. 129 22. 3.4 FUNCTIONS OF RANDOM VARIABLES - p. 132 23. EXERCISES 3.4 - p. 137 24. 3.5 LIMIT THEOREMS - p. 139 25. EXERCISES 3.5 - p. 145 26. 3.6 CHAPTER SUMMARY - p. 146 27. 3.7 COMPUTER EXAMPLES (OPTIONAL) - p. 147 28. PROJECTS FOR CHAPTER 3 - p. 151 29. EXERCISE 3B - p. 151 30. EXERCISE 3F - p. 152 31. Chapter 4. Sampling Probability Distributions - p. 153 32. Chapter 5. Statistical Estimation - p. 183 33. Chapter outline - p. 183 34. 5.1 INTRODUCTION - p. 184 35. 5.2 THE METHODS OF FINDING POINT ESTIMATORS - p. 185 36. EXERCISES 5.2 - p. 203 37. 5.3 SOME DESIRABLE PROPERTIES OF POINT ESTIMATORS - p. 207 38. EXERCISES 5.3 - p. 220 39. 5.4 A METHOD OF FINDING THE CONFIDENCE INTERVAL: PIVOTAL METHOD - p. 222 40. EXERCISES 5.4 - p. 228 41. 5.5 ONE-SAMPLE CONFIDENCE INTERVALS - p. 229 42. EXERCISES 5.5 - p. 237 43. 5.6 A CONFIDENCE INTERVAL FOR THE POPULATION VARIANCE - p. 242 44. EXERCISES 5.6 - p. 244 45. 5.7 CONFIDENCE INTERVAL CONCERNING TWO POPULATION PARAMETERS - p. 246 46. EXERCISES 5.7 - p. 251 47. 5.8 CHAPTER SUMMARY - p. 253 48. 5.9 COMPUTER EXAMPLES - p. 254 49. EXERCISES 5.9 - p. 257 50. 5.10 PROJECTS FOR CHAPTER 5 - p. 257 51. Chapter 6. Hypothesis Testing - p. 263 52. Chapter outline - p. 263 53. 6.1 INTRODUCTION - p. 263 54. 6.1.1 Sample Size - p. 270 55. EXERCISES 6.1 - p. 272 56. 6.2 THE NEYMAN–PEARSON LEMMA - p. 272 57. EXERCISES 6.2 - p. 278 58. 6.3 LIKELIHOOD RATIO TESTS - p. 278 59. EXERCISES 6.3 - p. 282 60. 6.4 HYPOTHESES FOR A SINGLE PARAMETER - p. 282 61. EXERCISES 6.4 - p. 290 62. 6.5 TESTING OF HYPOTHESES FOR TWO SAMPLES - p. 292 63. Equal Variances - p. 293 64. Unequal Variances: Welch’s t-Test (σ21 ≠ σ22) - p. 294 65. EXERCISES 6.5 - p. 301 66. 6.6 CHAPTER SUMMARY - p. 303 67. 6.7 COMPUTER EXAMPLES - p. 304 68. PROJECTS FOR CHAPTER 6 - p. 311 69. Chapter 7. Linear Regression Models - p. 313 70. Chapter outline - p. 313 71. 7.1 INTRODUCTION - p. 314 72. 7.2 THE SIMPLE LINEAR REGRESSION MODEL - p. 314 73. EXERCISES 7.2 - p. 324 74. 7.3 INFERENCES ON THE LEAST-SQUARES ESTIMATORS - p. 327 75. EXERCISES 7.3 - p. 333 76. 7.4 PREDICTING A PARTICULAR VALUE OF Y - p. 334 77. EXERCISES 7.4 - p. 336 78. 7.5 CORRELATION ANALYSIS - p. 337 79. EXERCISES 7.5 - p. 339 80. 7.6 MATRIX NOTATION FOR LINEAR REGRESSION - p. 340 81. EXERCISES 7.6 - p. 344 82. 7.7 REGRESSION DIAGNOSTICS - p. 345 83. 7.8 CHAPTER SUMMARY - p. 347 84. 7.9 COMPUTER EXAMPLES - p. 348 85. PROJECTS FOR CHAPTER 7 - p. 353 86. Chapter 8. Design of Experiments - p. 355 87. Chapter 9. Analysis of Variance - p. 383 88. Chapter 10. Bayesian Estimation and Inference - p. 429 89. Chapter outline - p. 429 90. 10.1 INTRODUCTION - p. 430 91. 10.2 BAYESIAN POINT ESTIMATION - p. 431 92. EXERCISES 10.2 - p. 444 93. 10.3 BAYESIAN CONFIDENCE INTERVAL OR CREDIBLE INTERVALS - p. 445 94. EXERCISES 10.3 - p. 448 95. 10.4 BAYESIAN HYPOTHESIS TESTING - p. 450 96. EXERCISES 10.4 - p. 452 97. 10.5 BAYESIAN DECISION THEORY - p. 453 98. EXERCISES 10.5 - p. 457 99. 10.6 EMPIRICAL BAYES ESTIMATES - p. 459 100. EXERCISES 10.6 - p. 471 101. 10.7 CHAPTER SUMMARY - p. 473 102. 10.8 COMPUTER EXAMPLES - p. 473 103. PROJECTS FOR CHAPTER 10 - p. 476 104. Chapter 11. Categorical Data Analysis and Goodness of Fit Tests and Applications - p. 479 105. Chapter 12. Nonparametric Statistics - p. 511 106. Chapter 13. Empirical Methods - p. 555 107. Chapter outline - p. 555 108. 13.1 INTRODUCTION - p. 556 109. 13.2 THE JACKKNIFE METHOD - p. 556 110. EXERCISES 13.2 - p. 560 111. 13.3 AN INTRODUCTION TO BOOTSTRAP METHODS - p. 561 112. EXERCISES 13.3 - p. 566 113. 13.4 THE EXPECTATION–MAXIMIZATION ALGORITHM - p. 566 114. EXERCISES 13.4 - p. 576 115. 13.5 INTRODUCTION TO MARKOV CHAIN MONTE CARLO - p. 576 116. EXERCISES 13.5 - p. 589 117. 13.6 CHAPTER SUMMARY - p. 590 118. 13.7 COMPUTER EXAMPLES - p. 591 119. 13.8 PROJECTS FOR CHAPTER 13 - p. 597 120. Chapter 14. Extreme Value Theory With Applications - p. 599 121. Chapter 15. Modeling Issues in Statistical Applications: An Overview - p. 633 122. Appendix A. Set Theory - p. 681 123. Appendix B. Review of Markov Chains - p. 685 124. Appendix C. Common Probability Distributions - p. 689 125. Appendix D. What is R? - p. 691 126. Appendix E. Probability Tables - p. 697 127. Bibliography - p. 733 128. References - p. 733 129. Index - p. 737
Mathematical Statistics with Applications in R, Fourth Edition offers a modern calculus-based theoretical introduction to mathematical statistics and applications that spans numerous foundational and essential concepts in the field. The book covers many modern statistical computational and simulation concepts, including Exploratory Data Analysis, the Jackknife, bootstrap methods, the EM algorithms, and Markov chain Monte Carlo (MCMC) methods such as the Metropolis algorithm, Metropolis-Hastings algorithm and the Gibbs sampler. The final chapter of the book provides a step-by-step approach to modelling, analysis, and interpretation data from real-world applications, from the environment and cyber security to health and finance.
By combining discussion on the theory of statistics with a wealth of engaging, real-world applications, this book helps students approach statistical problem-solving in a logical manner with accessible, step-by-step procedures on relatable topics. Computational aspects are covered through R and SAS examples.
1. 2.1 INTRODUCTION - p. 47 2. 2.2 RANDOM EVENTS AND PROBABILITY - p. 48 3. EXERCISES 2.2 - p. 52 4. 2.3 COUNTING TECHNIQUES AND CALCULATION OF PROBABILITIES - p. 55 5. EXERCISES 2.3 - p. 60 6. 2.4 CONDITIONAL PROBABILITY, INDEPENDENCE, AND BAYES’ RULE - p. 61 7. EXERCISES 2.4 - p. 67 8. 2.5 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS - p. 71 9. EXERCISES 2.5 - p. 77 10. 2.6 MOMENTS AND MOMENT-GENERATING FUNCTIONS - p. 79 11. EXERCISES 2.6 - p. 90 12. 2.7 CHAPTER SUMMARY - p. 92 13. 2.8 COMPUTER EXAMPLES (OPTIONAL) - p. 93 14. PROJECTS FOR CHAPTER 2 - p. 96 15. Chapter 3. Probability Distributions - p. 99 16. Chapter outline - p. 99 17. 3.1 INTRODUCTION - p. 100 18. 3.2 SPECIAL DISTRIBUTION FUNCTIONS - p. 100 19. EXERCISES 3.2 - p. 117 20. 3.3 JOINT PROBABILITY DISTRIBUTIONS - p. 121 21. EXERCISES 3.3 - p. 129 22. 3.4 FUNCTIONS OF RANDOM VARIABLES - p. 132 23. EXERCISES 3.4 - p. 137 24. 3.5 LIMIT THEOREMS - p. 139 25. EXERCISES 3.5 - p. 145 26. 3.6 CHAPTER SUMMARY - p. 146 27. 3.7 COMPUTER EXAMPLES (OPTIONAL) - p. 147 28. PROJECTS FOR CHAPTER 3 - p. 151 29. EXERCISE 3B - p. 151 30. EXERCISE 3F - p. 152 31. Chapter 4. Sampling Probability Distributions - p. 153 32. Chapter 5. Statistical Estimation - p. 183 33. Chapter outline - p. 183 34. 5.1 INTRODUCTION - p. 184 35. 5.2 THE METHODS OF FINDING POINT ESTIMATORS - p. 185 36. EXERCISES 5.2 - p. 203 37. 5.3 SOME DESIRABLE PROPERTIES OF POINT ESTIMATORS - p. 207 38. EXERCISES 5.3 - p. 220 39. 5.4 A METHOD OF FINDING THE CONFIDENCE INTERVAL: PIVOTAL METHOD - p. 222 40. EXERCISES 5.4 - p. 228 41. 5.5 ONE-SAMPLE CONFIDENCE INTERVALS - p. 229 42. EXERCISES 5.5 - p. 237 43. 5.6 A CONFIDENCE INTERVAL FOR THE POPULATION VARIANCE - p. 242 44. EXERCISES 5.6 - p. 244 45. 5.7 CONFIDENCE INTERVAL CONCERNING TWO POPULATION PARAMETERS - p. 246 46. EXERCISES 5.7 - p. 251 47. 5.8 CHAPTER SUMMARY - p. 253 48. 5.9 COMPUTER EXAMPLES - p. 254 49. EXERCISES 5.9 - p. 257 50. 5.10 PROJECTS FOR CHAPTER 5 - p. 257 51. Chapter 6. Hypothesis Testing - p. 263 52. Chapter outline - p. 263 53. 6.1 INTRODUCTION - p. 263 54. 6.1.1 Sample Size - p. 270 55. EXERCISES 6.1 - p. 272 56. 6.2 THE NEYMAN–PEARSON LEMMA - p. 272 57. EXERCISES 6.2 - p. 278 58. 6.3 LIKELIHOOD RATIO TESTS - p. 278 59. EXERCISES 6.3 - p. 282 60. 6.4 HYPOTHESES FOR A SINGLE PARAMETER - p. 282 61. EXERCISES 6.4 - p. 290 62. 6.5 TESTING OF HYPOTHESES FOR TWO SAMPLES - p. 292 63. Equal Variances - p. 293 64. Unequal Variances: Welch’s t-Test (σ21 ≠ σ22) - p. 294 65. EXERCISES 6.5 - p. 301 66. 6.6 CHAPTER SUMMARY - p. 303 67. 6.7 COMPUTER EXAMPLES - p. 304 68. PROJECTS FOR CHAPTER 6 - p. 311 69. Chapter 7. Linear Regression Models - p. 313 70. Chapter outline - p. 313 71. 7.1 INTRODUCTION - p. 314 72. 7.2 THE SIMPLE LINEAR REGRESSION MODEL - p. 314 73. EXERCISES 7.2 - p. 324 74. 7.3 INFERENCES ON THE LEAST-SQUARES ESTIMATORS - p. 327 75. EXERCISES 7.3 - p. 333 76. 7.4 PREDICTING A PARTICULAR VALUE OF Y - p. 334 77. EXERCISES 7.4 - p. 336 78. 7.5 CORRELATION ANALYSIS - p. 337 79. EXERCISES 7.5 - p. 339 80. 7.6 MATRIX NOTATION FOR LINEAR REGRESSION - p. 340 81. EXERCISES 7.6 - p. 344 82. 7.7 REGRESSION DIAGNOSTICS - p. 345 83. 7.8 CHAPTER SUMMARY - p. 347 84. 7.9 COMPUTER EXAMPLES - p. 348 85. PROJECTS FOR CHAPTER 7 - p. 353 86. Chapter 8. Design of Experiments - p. 355 87. Chapter 9. Analysis of Variance - p. 383 88. Chapter 10. Bayesian Estimation and Inference - p. 429 89. Chapter outline - p. 429 90. 10.1 INTRODUCTION - p. 430 91. 10.2 BAYESIAN POINT ESTIMATION - p. 431 92. EXERCISES 10.2 - p. 444 93. 10.3 BAYESIAN CONFIDENCE INTERVAL OR CREDIBLE INTERVALS - p. 445 94. EXERCISES 10.3 - p. 448 95. 10.4 BAYESIAN HYPOTHESIS TESTING - p. 450 96. EXERCISES 10.4 - p. 452 97. 10.5 BAYESIAN DECISION THEORY - p. 453 98. EXERCISES 10.5 - p. 457 99. 10.6 EMPIRICAL BAYES ESTIMATES - p. 459 100. EXERCISES 10.6 - p. 471 101. 10.7 CHAPTER SUMMARY - p. 473 102. 10.8 COMPUTER EXAMPLES - p. 473 103. PROJECTS FOR CHAPTER 10 - p. 476 104. Chapter 11. Categorical Data Analysis and Goodness of Fit Tests and Applications - p. 479 105. Chapter 12. Nonparametric Statistics - p. 511 106. Chapter 13. Empirical Methods - p. 555 107. Chapter outline - p. 555 108. 13.1 INTRODUCTION - p. 556 109. 13.2 THE JACKKNIFE METHOD - p. 556 110. EXERCISES 13.2 - p. 560 111. 13.3 AN INTRODUCTION TO BOOTSTRAP METHODS - p. 561 112. EXERCISES 13.3 - p. 566 113. 13.4 THE EXPECTATION–MAXIMIZATION ALGORITHM - p. 566 114. EXERCISES 13.4 - p. 576 115. 13.5 INTRODUCTION TO MARKOV CHAIN MONTE CARLO - p. 576 116. EXERCISES 13.5 - p. 589 117. 13.6 CHAPTER SUMMARY - p. 590 118. 13.7 COMPUTER EXAMPLES - p. 591 119. 13.8 PROJECTS FOR CHAPTER 13 - p. 597 120. Chapter 14. Extreme Value Theory With Applications - p. 599 121. Chapter 15. Modeling Issues in Statistical Applications: An Overview - p. 633 122. Appendix A. Set Theory - p. 681 123. Appendix B. Review of Markov Chains - p. 685 124. Appendix C. Common Probability Distributions - p. 689 125. Appendix D. What is R? - p. 691 126. Appendix E. Probability Tables - p. 697 127. Bibliography - p. 733 128. References - p. 733 129. Index - p. 737
