CHAPTER 1 Experiments—Decision Spaces 1
1 Introduction 1
2 Vector Lattices—L-Spaces—Transitions 3
3 Experiments—Decision Procedures 5
4 A Basic Density Theorem 6
5 Building Experiments from Other Ones 10
6 Representations—Markov Kernels 11
CHAPTER 2 Some Results from Decision Theory:Deficiencies 16
1 Introduction 16
2 Characterization of the Spaces of Risk Functions:Minimax Theorem 16
3 Deficiencies;Distances 18
4 The Form of Bayes Risks—Choquet Lattices 23
CHAPTER 3 Likelihood Ratios and Conical Measures 29
1 Introduction 29
2 Homogeneous Functions of Measures 30
3 Deficiencies for Binary Experiments:Isometries 34
4 Weak Convergence of Experiments 37
5 Boundedly Complete Experiments 40
6 Convolutions:Hellinger Transforms 42
7 The Blackwell-Sherman-Stein Theorem 43
CHAPTER 4 Some Basic Inequalities 46
1 Introduction 46
2 Hellinger Distances:L1-Norm 46
3 Approximation Properties for Likelihood Ratios 49
4 Inequalities for Conditional Distributions 52
CHAPTER 5 Sufficiency and Insufficiency 57
1 Introduction 57
2 Projections and Conditional Expectations 58
3 Equivalent Definitions for Sufficiency 62
4 Insufficiency 67
5 Estimating Conditional Distributions 73
CHAPTER 6 Domination,Compactness,Contiguity 81
1 Introduction 81
2 Definitions and Elementary Relations 81
3 Contiguity 84
4 Strong Compactness and a Result of D.Lindae 92
CHAPTER 7 Some Limit Theorems 96
1 Introduction 96
2 Convergence in Distribution or in Probability 97
3 Distinguished Sequences of Statistics 99
4 Lower-Semicontinuity for Spaces of Risk Functions 108
5 A Result on Asymptotic Admissibility 112
CHAPTER 8 Invariance Properties 118
1 Introduction 118
2 The Markov-Kakutani Fixed Point Theorem 119
3 A Lifting Theorem and Some Applications 125
4 Automatic Invariance of Limits 132
5 Invariant Exponential Families 144
6 The Hunt-Stein Theorem and Related Results 151
CHAPTER 9 Infinitely Divisible,Gaussian,and Poisson Experiments 154
1 Introduction 154
2 Infinite Divisibility 154
3 Gaussian Experiments 155
4 Poisson Experiments 159
5 A Central Limit Theorem 165
CHAPTER 10 Asymptotically Gaussian Experiments:Local Theory 172
1 Introduction 172
2 Convergence to a Gaussian Shift Experiment 173
3 A Framework which Arises in Many Applications 179
4 Weak Convergence of Distributions 184
5 An Application of a Martingale Limit Theorem 187
6 Asymptotic Admissibility and Minimaxity 195
CHAPTER 11 Asymptotic Normality—Global 206
1 Introduction 206
2 Preliminary Explanations 208
3 Construction of Centering Variables 213
4 Definitions Relative to Quadratic Approximations 219
5 Asymptotic Properties of the Centerings ？ 225
6 The Asymptotically Gaussian Case 238
7 Some Particular Cases 268
8 Reduction to the Gaussian Case by Small Distortions 283
9 The Standard Tests and Confidence Sets 293
10 Minimum x2 and Relatives 305
CHAPTER 12 Posterior Distributions and Bayes Solutions 324
1 Introduction 324
2 Inequalities on Conditional Distributions 325
3 Asymptotic Behavior of Bayes Procedures 330
4 Approximately Gaussian Posterior Distributions 336
CHAPTER 13 An Approximation Theorem for Certain Sequential Experiments 346
1 Introduction 346
2 Notations and Assumptions 347
3 Basic Auxiliary Lemmas 350
4 Reduction Theorems 354
5 Remarks on Possible Applications 362
CHAPTER 14 Approximation by Exponential Families 370
1 Introduction 370
2 A Lemma on Approximate Sufficiency 371
3 Homogeneous Experiments of Finite Rank 377
4 Approximation by Experiments of Finite Rank 387
5 Construction of Distinguished Sequences of Estimates 391
CHAPTER 15 Sums of Independent Random Variables 399
1 Introduction 399
2 Concentration Inequalities 401
3 Compactness and Shift-Compactness 419
4 Poisson Exponentials and Approximation Theorems 423
5 Limit Theorems and Related Results 434
6 Sums of Independent Stochastic Processes 444
CHAPTER 16 Independent Observations 457
1 Introduction 457
2 Limiting Distributions for Likelihood Ratios 458
3 Conditions for Asymptotic Normality 468
4 Tests and Distances 475
5 Estimates for Finite Dimensional Parameter Spaces 493
6 The Risk of Formal Bayes Procedures 509
7 Empirical Measures and Cumulatives 529
8 Empirical Measures on Vapnik-？ervonenkis Classes 541
CHAPTER 17 Independent Identically Distributed Observations 555
1 Introduction 555
2 Hilbert Spaces Around a Point 556
3 A Special Role for ？ Differentiability in Quadratic Mean 573
4 Asymptotic Normality for Rates Other than ？ 590
5 Existence of Consistent Estimates 594
6 Estimates Converging at the ？- Rate 604
7 The Behavior of Posterior Distributions 614
8 Maximum Likelihood 621
9 Some Cases where the Number of Observations Is Random 625
Appendix:Results from Classical Analysis 634
1 The Language of Set Theory 634
2 Topological Spaces 638
3 Uniform Spaces 640
4 Metric Spaces 641
5 Spaces of Functions 643
6 Vector Spaces 645
7 Vector Lattices 650
8 Vector Lattices Arising from Experiments 657
9 Lattices of Numerical Functions 672
10 Extensions of Positive Linear Functions 677
11 Smooth Linear Functionals 697
12 Derivatives and Tangents 707
Bibliography 727
Index 737