Statistical Decision Theory: Estimation, Testing, and Selection Book

Preface     VI
Statistical Models     1
Exponential Families     2
Priors and Conjugate Priors for Exponential Families     16
Divergences in Binary Models     31
Information in Bayes Models     52
L[subscript 2]-Differentiability, Fisher Information     58
Solutions to Selected Problems     67
Tests in Models with Monotonicity Properties     75
Stochastic Ordering and Monotone Likelihood Ratio     75
Tests in Binary Models and Models with MLR     83
Solutions to Selected Problems     100
Statistical Decision Theory     104
Decisions in Statistical Models     104
Convergence of Decisions     114
Continuity Properties of the Risk     118
Minimum Average Risk, Bayes Risk, Posterior Risk     121
Bayes and Minimax Decisions     133
[Gamma]-Minimax Decisions     141
Minimax Theorem     146
Complete Classes     149
Solutions to Selected Problems     153
Comparison of Models, Reduction by Sufficiency     156
Comparison and Randomization of Models     156
Comparison of Finite Models by Standard Distributions     166
Sufficiency in Dominated Models     177
Completeness, Ancillarity, and Minimal Sufficiency     188
Solutions to Selected Problems     194
Invariant Statistical Decision Models     198
Invariant Models and Invariant Statistics     198
Invariant Decision Problems     204
Hunt-Stein Theorem     213
Equivariant Estimators, Girshick-Savage Theorem     222
Solutions to Selected Problems     232
Large Sample Approximations of Models and Decisions     235
Distances of Statistical Models     235
Convergence of Models     241
Weak Convergence of Binary Models     248
Asymptotically Normal Models     265
Gaussian Models     266
The LAN and ULAN Property     269
Asymptotic Lower Risk Bounds, Hajek-LeCam Bound     281
Solutions to Selected Problems     287
Estimation     293
Lower Information Bounds in Estimation Problems     293
Unbiased Estimators with Minimal Risk     301
Bayes and Generalized Bayes Estimators     309
Admissibility of Estimators, Shrinkage Estimators     315
Consistency of Estimators     319
Consistency of M-Estimators and MLEs     319
Consistency in Bayes Models     347
Asymptotic Distributions of Estimators     359
Asymptotic Distributions of M-Estimators     359
Asymptotic Distributions of MLEs     374
Asymptotic Normality of the Posterior     379
Local Asymptotic Optimality of MLEs     386
Solutions to Selected Problems     400
Testing     406
Best Tests for Exponential Families     406
Tests for One-Parameter Exponential Families     406
Tests in Multivariate Normal Distributions     417
Tests for d-Parameter Exponential Families     420
Confidence Regions and Confidence Bounds     431
Bayes Tests     437
Uniformly Best Invariant Tests     443
Exponential Rates of Error Probabilities     450
U-Statistics and Rank Statistics     454
Statistics with Estimated Parameters     470
Asymptotic Null Distribution     473
Locally Asymptotically Optimal Tests     485
Testing of Univariate Parameters     485
Testing of Multivariate Parameters     503
Solutions to Selected Problems     510
Selection     516
The Selection Models      516
Optimal Point Selections     520
Point Selections, Loss, and Risk     520
Point Selections in Balanced Models     528
Point Selections in Unbalanced Models     536
Point Selections with Estimation     542
Optimal Subset Selections     547
Subset Selections, Loss, and Risk     547
[Gamma]-Minimax Subset Selections     556
Optimal Multistage Selections     561
Common Sample Size per Stage and Hard Elimination     561
Bayes Sampling Designs for Adaptive Sampling     582
Asymptotically Optimal Point Selections     587
Exponential Rate of Error Probabilities     587
Locally Asymptotically Optimal Point Selections     592
Rank Selection Rules     607
Solutions to Selected Problems     609
Appendix: Topics from Analysis, Measure Theory, and Probability Theory     615
Topics from Analysis     615
Topics from Measure Theory     617
Topics from Probability Theory     623
Appendix: Common Notation and Distributions     631
Common Notation     631
Common Distributions     635
References     640
Author Index      663
Subject Index     668