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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