Mathematical Statistics

MATHEMATICAL STATISTICS......Page 4
Contents......Page 7
Preface......Page 10
Table of Contents......Page 0
Operations on events......Page 13
1.2 Probability measures......Page 14
Consequences of the axioms......Page 15
Finite sample spaces......Page 17
1.3 Conditional probability and independence......Page 21
Bayes’ Theorem......Page 23
Independence......Page 26
1.4 Random variables......Page 28
Probabilitydistributions......Page 29
Discrete random variables......Page 31
Continuous random variables......Page 33
Hazard functions......Page 39
1.5 Transformations of random variables......Page 41
1.6 Expected values......Page 43
Variance and moment generating function......Page 51
1.7 Problems and complements......Page 59
2.1 Introduction......Page 67
2.2 Discrete and continuous random vectors......Page 68
Independent random variables......Page 71
Transformations......Page 73
Expected values......Page 78
Covariance and Correlation......Page 80
2.3 Conditional distributions and expected values......Page 86
The Multivariate Normal Distribution......Page 92
The X2 distribution and orthogonal transformations......Page 93
The t and F distributions......Page 97
Projection matrices......Page 98
2.5 Poisson processes......Page 102
2.6 Generating random variables......Page 107
2.7 Problems and complements......Page 111
3.1 Introduction......Page 125
3.2 Convergence in probability and distribution......Page 127
Some important results......Page 129
3.3 Weak Law of Large Numbers......Page 134
Proving the WLLN......Page 135
3.4 Proving convergence in distribution......Page 137
3.5 Central Limit Theorems......Page 144
Proving the CLT......Page 145
Using the CLT as an approximation theorem......Page 148
Some other Central Limit Theorems......Page 153
Multivariate Central Limit Theorem......Page 158
Variance stabilizing transformations......Page 161
A CLT for dependent random variables......Page 163
Monte Carlo integration.......Page 165
3.7 Convergence with probability 1......Page 170
3.8 Problems and complements......Page 174
4.1 Introduction......Page 184
4.2 Statistical models......Page 185
Exponential families......Page 188
Statistics......Page 190
4.3 Sufficiency......Page 192
4.4 Point estimation......Page 197
4.5 The substitution principle......Page 201
The substitution principle......Page 203
Method of moments......Page 206
4.6 Influence curves......Page 209
4.7 Standard errors and their estimation......Page 218
4.8 Asymptotic relative efficiency......Page 222
4.9 The jackknife......Page 226
The jackknife estimator of bias......Page 228
The jackknife estimator of variance......Page 232
Comparing the jackknife and Delta Method estimators......Page 235
4.10 Problems and complements......Page 236
5.2 The likelihood function......Page 248
5.3 The likelihood principle......Page 253
5.4 Asymptotic theory for MLEs......Page 255
Estimating standard errors......Page 265
Multiparameter models......Page 267
5.5 Misspecified models......Page 271
5.6 Non-parametric maximum likelihood estimation......Page 279
The Newton-Raphson algorithm......Page 280
The Fisher scoring algorithm......Page 285
The EM algorithm......Page 286
Comparing the Newton-Raphson and EM algorithms......Page 292
5.8 Bayesian estimation......Page 293
Conjugate and ignorance priors......Page 298
5.9 Problems and complements......Page 303
6.2 Decision theory......Page 317
6.3 Minimum variance unbiased estimation......Page 321
Complete and Sufficient Statistics......Page 326
6.4 The Cramèr-Rao lower bound......Page 332
6.5 Asymptotic efficiency......Page 336
6.6 Problems and complements......Page 342
7.1 Confidence intervals and regions......Page 349
Pivotal method......Page 351
Confidence regions......Page 357
7.2 Highest posterior density regions......Page 360
7.3 Hypothesis testing......Page 364
Uniformly most powerful tests......Page 371
Other most powerful tests......Page 374
7.4 Likelihood ratio tests......Page 381
Asymptotic distribution of the LR statistic......Page 383
Other likelihood based tests......Page 386
P-values......Page 391
Obtaining confidence regions from hypothesis tests......Page 393
Confidence intervals and tests based on non-parametric likelihood......Page 396
7.6 Problems and complements......Page 398
8.1 Linear models......Page 412
8.2 Estimation in linear models......Page 413
8.3 Hypothesis testing in linear models......Page 417
Power of the F test......Page 420
8.4 Non-normal errors......Page 425
Some large sample theory......Page 427
Other estimation methods......Page 429
8.5 Generalized linear models......Page 432
Likelihood functions and estimation......Page 433
Inference for generalized linear models......Page 435
Numerical computation of parameter estimates......Page 436
8.6 Quasi-Likelihood models......Page 438
Estimation of the dispersion parameter......Page 440
8.7 Problems and complements......Page 441
9.1 Introduction......Page 453
9.2 Tests based on the Multinomial distribution......Page 454
Log-linear models......Page 460
9.3 Smooth goodness-of-fit tests......Page 463
Tests based on probability plots......Page 467
9.4 Problems and complements......Page 471
References......Page 477