Mathematical Statistics - A Unified Introduction

Contents......Page 11
1.1 Introduction......Page 30
1.2.1 Plotting Data......Page 31
1.2.2 Location Models......Page 32
1.3.1 Data from Several Treatments......Page 33
1.3.2 Centered Models......Page 35
1.3.3 Degrees of Freedom......Page 36
1.4.1 Cross-Classified Observations......Page 37
1.4.2 Additive Models......Page 39
1.4.3 Balanced Designs......Page 40
1.4.4 Interaction......Page 42
1.4.5 Centering Full Models......Page 43
1.5.1 Interpolating Between Levels......Page 44
1.5.2 Simple Linear Regression......Page 46
1.6.1 Double Interpolation......Page 48
1.6.2 Multiple Linear Regression......Page 50
1.7.1 Counted Data......Page 51
1.7.2 Independence Models......Page 53
1.7.3 Loglinear Models......Page 54
1.7.4 Loglinear Independence Models......Page 55
1.7.5 Loglinear Saturated Models*......Page 57
1.8.1 Interpolating in Contingency Tables......Page 58
1.8.2 Linear Logistic Regression......Page 60
1.9 Summary......Page 61
1.10 Exercises......Page 62
1.11 Supplementary Exercises......Page 66
2.1 Introduction......Page 72
2.2.1 Multiple Observations as Vectors......Page 73
2.2.2 Distances as Errors......Page 75
2.3.1 Simple Proportion Models......Page 76
2.3.2 Estimating the Constant......Page 78
2.3.3 Solving the Problem Using Matrix Notation......Page 80
2.3.4 Geometric Degrees of Freedom......Page 81
2.3.5 Schwarz's Inequality......Page 82
2.4.1 Least-Squares Location Estimation......Page 83
2.4.2 Sample Variance......Page 84
2.5.1 Analysis of Variance......Page 85
2.5.2 Geometric Interpretation......Page 87
2.5.3 ANOVA Tables......Page 89
2.5.4 The F-Statistic......Page 90
2.5.5 The Kruskal–Wallis Statistic......Page 92
2.6.1 Estimates for Simple Linear Regression......Page 93
2.6.2 ANOVA for Regression......Page 95
2.7.1 Standardizing the Regression Line......Page 96
2.7.2 Properties of the Sample Correlation......Page 97
2.8.1 ANOVA for Two-Way Layouts......Page 99
2.8.2 Additive Models......Page 101
2.10 Exercises......Page 103
2.11 Supplementary Exercises......Page 106
3.1 Introduction......Page 110
3.2.1 What Is Probability?......Page 111
3.2.2 Probabilities by Counting......Page 112
3.3.1 Basic Rules for Counting......Page 114
3.3.2 Counting Lists......Page 115
3.3.3 Combinations......Page 117
3.3.4 Multinomial Counting......Page 118
3.4.1 Complicated Counts......Page 119
3.4.2 The Birthday Problem......Page 120
3.4.3 General Principles About Probability......Page 121
3.5.1 An Upper Bound......Page 123
3.5.2 A Lower Bound......Page 125
3.5.3 A Useful Approximation......Page 126
3.6 Sampling......Page 127
3.8 Exercises......Page 128
3.9 Supplementary Exercises......Page 131
4.1 Introduction......Page 136
4.2.1 Uniform Geometric Probability......Page 137
4.2.2 General Properties......Page 139
4.3.2 Rules for Combining Events......Page 140
4.4.1 In General......Page 141
4.4.2 Axioms of Probability......Page 142
4.4.3 Consequences of the Axioms......Page 143
4.5.1 Definition......Page 144
4.5.2 Examples......Page 145
4.6.1 Partitions......Page 146
4.6.2 Division into Cases......Page 147
4.6.3 Bayes's Theorem......Page 149
4.6.4 Bayes's Theorem Applied to Partitions......Page 150
4.7.1 Irrelevant Conditions......Page 151
4.7.3 Near-Independence......Page 152
4.8.1 Probability Density......Page 153
4.8.2 Sigma Algebras and Borel Algebras*......Page 156
4.8.3 Kolmogorov's Axiom*......Page 158
4.10 Exercises......Page 161
4.11 Supplementary Exercises......Page 163
5.1 Introduction......Page 166
5.2.1 Some Simple Examples......Page 167
5.2.2 Discrete Random Variables......Page 168
5.2.3 The Negative Hypergeometric Family......Page 169
5.3.1 The Hypergeometric Family......Page 171
5.3.3 Fisher's Test for Independence......Page 173
5.3.5 The Sign Test......Page 175
5.4.1 Some Properties......Page 176
5.4.2 Continuous Variables......Page 177
5.4.3 Symmetry and Duality......Page 179
5.5.1 Average Values......Page 181
5.5.2 Discrete Random Variables......Page 182
5.5.3 The Method of Indicators......Page 183
5.6.2 Compatibility with the Data......Page 185
5.7 Summary......Page 187
5.8 Exercises......Page 188
5.9 Supplementary Exercises......Page 192
6.1 Introduction......Page 196
6.2.1 The Geometric Approximation......Page 197
6.2.3 Negative Binomial Approximations......Page 198
6.2.4 Negative Binomial Variables......Page 199
6.2.5 Convergence in Distribution......Page 200
6.3.1 Binomial Approximations......Page 201
6.3.2 Binomial Random Variables......Page 202
6.3.3 Bernoulli Processes......Page 204
6.4.1 Poisson Approximation to Binomial Probabilities......Page 205
6.4.2 Approximation to the Negative Binomial......Page 206
6.4.3 Poisson Random Variables......Page 207
6.5 More About Expectation......Page 208
6.6.1 Expectations of Functions......Page 211
6.6.2 Variance......Page 213
6.6.3 Variances of Some Families......Page 214
6.7.1 Estimating Binomial p......Page 216
6.7.2 Confidence Bounds for Binomial p......Page 217
6.7.3 Confidence Intervals......Page 218
6.7.4 Two-Sided Hypothesis Tests......Page 219
6.8 The Poisson Limit of the Negative Hypergeometric Family*......Page 220
6.9 Summary......Page 222
6.10 Exercises......Page 223
6.11 Supplementary Exercises......Page 227
7.1 Introduction......Page 230
7.2.1 Multinomial Random Vectors......Page 231
7.2.2 Marginal and Conditional Distributions......Page 232
7.3.1 Random Coordinates......Page 235
7.3.2 Multivariate Cumulative Distribution Functions......Page 237
7.4.1 Independence and Random Samples......Page 239
7.4.2 Sums of Random Vectors......Page 240
7.4.3 Convolutions......Page 241
7.5.2 Conditional Expectations......Page 242
7.5.3 Regression......Page 243
7.5.4 Linear Regression......Page 244
7.5.5 Covariance......Page 246
7.5.6 The Correlation Coefficient......Page 247
7.6.1 Expectations and Variances......Page 248
7.6.2 The Covariance Matrix......Page 249
7.6.4 Statistical Properties of Sample Means and Variances......Page 250
7.6.5 The Method of Indicators......Page 252
7.7.2 Markov's Inequality......Page 254
7.7.3 Convergence in Mean Squared Error......Page 255
7.8.1 Parameters in Models as Random Variables......Page 256
7.8.2 An Example of Bayesian Inference......Page 257
7.9 Summary......Page 258
7.10 Exercises......Page 259
7.11 Supplementary Exercises......Page 263
8.1 Introduction......Page 266
8.2.1 Posterior Probability of a Parameter Value......Page 267
8.2.2 Maximum Likelihood......Page 268
8.3.1 Ratio of the Maximum Likelihood to a Hypothetical Likelihood......Page 270
8.3.2 G-Squared......Page 271
8.4.1 Chi-Squared......Page 272
8.4.2 Comparing the Two Statistics......Page 273
8.4.4 Multinomial Models......Page 274
8.5.1 Conditions for a Maximum......Page 275
8.5.2 Proportional Fitting......Page 277
8.5.3 Iterative Proportional Fitting*......Page 278
8.5.4 Why Does It Work?*......Page 281
8.6.1 Relative G-Squared......Page 282
8.6.2 An ANOVA-like Table......Page 283
8.7.2 General Logistic Regression......Page 285
8.8.1 Linear Approximation to a Root......Page 287
8.8.2 Dose–Response with Historical Controls......Page 288
8.9 Summary......Page 289
8.10 Exercises......Page 290
8.11 Supplementary Exercises......Page 292
9.1 Introduction......Page 296
9.2.2 Continuous Variables......Page 297
9.3.1 How Would It Look?......Page 298
9.3.2 How to Construct a Poisson Process......Page 299
9.3.3 Spacings Between Events......Page 301
9.3.4 Gamma Variables......Page 302
9.3.5 Poisson Process as the Limit of a Hypergeometric Process*......Page 303
9.4.1 Transforming Variables......Page 305
9.4.2 Gamma Densities......Page 306
9.4.3 General Properties......Page 307
9.4.4 Interpretation......Page 309
9.5.1 Order Statistics......Page 312
9.5.2 Dirichlet Processes......Page 313
9.5.3 Beta Variables......Page 314
9.5.4 Beta Densities......Page 316
9.5.5 Connections......Page 317
9.6.1 Hypothesis Tests and Parameter Estimates......Page 319
9.6.2 Confidence Intervals......Page 320
9.6.3 Inferences About the Shape Parameter......Page 321
9.7.1 Alternative Hypotheses......Page 322
9.7.2 Most Powerful Tests......Page 323
9.8 Summary......Page 325
9.9 Exercises......Page 326
9.10 Supplementary Exercises......Page 328
10.1 Introduction......Page 330
10.2.2 Quantile Functions in General......Page 331
10.2.3 Continuous Quantile Functions......Page 333
10.3.1 Expectation as the Integral of a Quantile Function......Page 334
10.3.2 Markov's Inequality Revisited......Page 337
10.4.1 Changing Variables in a Density......Page 338
10.4.2 Expectation in Terms of a Density......Page 339
10.5.1 Shape of a Gamma Density......Page 341
10.5.2 Quadratic Approximation to the Log-Density......Page 342
10.5.3 Standard Normal Density......Page 345
10.5.5 Approximate Gamma Probabilities......Page 347
10.5.6 Computing Normal Probabilities......Page 348
10.5.7 Normal Tail Probabilities......Page 349
10.6.1 Dual Probabilities......Page 350
10.6.2 Continuity Correction......Page 352
10.7.1 The Normal Family......Page 353
10.7.2 Approximate Poisson Intervals......Page 354
10.7.3 Approximate Gamma Intervals......Page 355
10.9 Exercises......Page 356
10.10 Supplementary Exercises......Page 359
11.1 Introduction......Page 362
11.2.2 The General Case......Page 363
11.3.1 Two Order Statistics at Once......Page 364
11.3.2 Joint Density of Two Order Statistics......Page 365
11.3.3 Joint Densities in General......Page 366
11.3.4 The Family of Divisions of an Interval......Page 367
11.4.1 Affine Multivariate Transformations......Page 368
11.4.2 Dirichlet Densities......Page 370
11.4.3 Some Properties of Dirichlet Variables......Page 371
11.4.4 General Change of Variables......Page 373
11.5.1 Gammas Conditioned on Their Sum......Page 374
11.5.3 Gamma Densities in General......Page 375
11.5.4 Chi-Squared Variables......Page 377
11.6.1 Bayes's Theorem Revisited......Page 378
11.6.2 Application to Gamma Observations......Page 379
11.7.2 Linear Combinations of Normal Variables......Page 381
11.7.4 Approximating a Beta Variable......Page 383
11.8.1 Binomial Variables with Large Variance......Page 384
11.8.2 Negative Binomial Variables with Small Coefficient of Variation......Page 385
11.9.1 Approximating Two Order Statistics......Page 386
11.9.2 Correlated Normal Variables......Page 387
11.10.1 Family Relationships......Page 388
11.10.2 Asymptotic Normality......Page 389
11.11 Summary......Page 390
11.12 Exercises......Page 391
11.13 Supplementary Exercises......Page 393
12.1 Introduction......Page 396
12.2.1 A Probability Model for Errors......Page 397
12.2.2 Statistics of Fit for the Error Model......Page 398
12.3.1 Independence Models for Errors......Page 399
12.3.2 Distribution of R-squared......Page 400
12.3.3 Elementary Errors......Page 401
12.4.1 Continuous Likelihoods......Page 402
12.4.2 Maximum Likelihood with Normal Errors......Page 403
12.4.3 Unbiased Variance Estimates......Page 404
12.5.1 When the Variance Is Known......Page 405
12.5.2 When the Variance Is Unknown......Page 406
12.6.1 Matrix Form......Page 407
12.6.2 Centered Form......Page 408
12.6.3 Least-Squares Estimates......Page 409
12.6.4 Homoscedastic Errors......Page 410
12.6.5 Linear Combinations of Parameters......Page 412
12.7.2 Gauss–Markov Theorem......Page 413
12.8.1 The Score Estimator......Page 414
12.8.2 How Good Is It?......Page 416
12.8.3 The Information Inequality......Page 417
12.9 Summary......Page 419
12.10 Exercises......Page 420
12.11 Supplementary Exercises......Page 421
13.1 Introduction......Page 424
13.2.2 The P.G.F. Representation......Page 425
13.2.3 The P.G.F. As an Expectation......Page 427
13.2.4 Applications to Compound Variables......Page 428
13.2.5 Factorial Moments......Page 430
13.3.1 Comparison with Exponential Variables......Page 431
13.3.2 The M.G.F. as an Expectation......Page 433
13.4.1 Poisson Limits......Page 434
13.4.2 Law of Large Numbers......Page 435
13.4.3 Normal Limits......Page 436
13.4.4 A Central Limit Theorem......Page 437
13.5.1 Natural Exponential Forms......Page 439
13.5.2 Expectations......Page 440
13.5.3 Natural Parameters......Page 441
13.5.5 Other Sufficient Statistics......Page 442
13.6.1 Conditional Improvement......Page 443
13.6.2 Sufficient Statistics......Page 445
13.7.1 Tail Probability Approximation......Page 446
13.7.2 Tilting a Random Variable......Page 447
13.7.3 Normal Tail Approximation......Page 448
13.7.4 Poisson Tail Approximations......Page 450
13.7.5 Small-Sample Asymptotics......Page 451
13.9 Exercises......Page 452
13.10 Supplementary Exercises......Page 455
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W......Page 474