统计决策理论中的渐近方法 英文版PDF|Epub|txt|kindle电子书版本网盘下载

统计决策理论中的渐近方法 英文版PDF格式电子书版下载

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CHAPTER 1 Experiments—Decision Spaces1

1 Introduction1

2 Vector Lattices—L-Spaces—Transitions3

3 Experiments—Decision Procedures5

4 A Basic Density Theorem6

5 Building Experiments from Other Ones10

6 Representations—Markov Kernels11

CHAPTER 2 Some Results from Decision Theory:Deficiencies16

1 Introduction16

2 Characterization of the Spaces of Risk Functions:Minimax Theorem16

3 Deficiencies;Distances18

4 The Form of Bayes Risks—Choquet Lattices23

CHAPTER 3 Likelihood Ratios and Conical Measures29

1 Introduction29

2 Homogeneous Functions of Measures30

3 Deficiencies for Binary Experiments:Isometries34

4 Weak Convergence of Experiments37

5 Boundedly Complete Experiments40

6 Convolutions:Hellinger Transforms42

7 The Blackwell-Sherman-Stein Theorem43

CHAPTER 4 Some Basic Inequalities46

1 Introduction46

2 Hellinger Distances:L1-Norm46

3 Approximation Properties for Likelihood Ratios49

4 Inequalities for Conditional Distributions52

CHAPTER 5 Sufficiency and Insufficiency57

1 Introduction57

2 Projections and Conditional Expectations58

3 Equivalent Definitions for Sufficiency62

4 Insufficiency67

5 Estimating Conditional Distributions73

CHAPTER 6 Domination,Compactness,Contiguity81

1 Introduction81

2 Definitions and Elementary Relations81

3 Contiguity84

4 Strong Compactness and a Result of D.Lindae92

CHAPTER 7 Some Limit Theorems96

1 Introduction96

2 Convergence in Distribution or in Probability97

3 Distinguished Sequences of Statistics99

4 Lower-Semicontinuity for Spaces of Risk Functions108

5 A Result on Asymptotic Admissibility112

CHAPTER 8 Invariance Properties118

1 Introduction118

2 The Markov-Kakutani Fixed Point Theorem119

3 A Lifting Theorem and Some Applications125

4 Automatic Invariance of Limits132

5 Invariant Exponential Families144

6 The Hunt-Stein Theorem and Related Results151

CHAPTER 9 Infinitely Divisible,Gaussian,and Poisson Experiments154

1 Introduction154

2 Infinite Divisibility154

3 Gaussian Experiments155

4 Poisson Experiments159

5 A Central Limit Theorem165

CHAPTER 10 Asymptotically Gaussian Experiments:Local Theory172

1 Introduction172

2 Convergence to a Gaussian Shift Experiment173

3 A Framework which Arises in Many Applications179

4 Weak Convergence of Distributions184

5 An Application of a Martingale Limit Theorem187

6 Asymptotic Admissibility and Minimaxity195

CHAPTER 11 Asymptotic Normality—Global206

1 Introduction206

2 Preliminary Explanations208

3 Construction of Centering Variables213

4 Definitions Relative to Quadratic Approximations219

5 Asymptotic Properties of the Centerings ？225

6 The Asymptotically Gaussian Case238

7 Some Particular Cases268

8 Reduction to the Gaussian Case by Small Distortions283

9 The Standard Tests and Confidence Sets293

10 Minimum x2 and Relatives305

CHAPTER 12 Posterior Distributions and Bayes Solutions324

1 Introduction324

2 Inequalities on Conditional Distributions325

3 Asymptotic Behavior of Bayes Procedures330

4 Approximately Gaussian Posterior Distributions336

CHAPTER 13 An Approximation Theorem for Certain Sequential Experiments346

1 Introduction346

2 Notations and Assumptions347

3 Basic Auxiliary Lemmas350

4 Reduction Theorems354

5 Remarks on Possible Applications362

CHAPTER 14 Approximation by Exponential Families370

1 Introduction370

2 A Lemma on Approximate Sufficiency371

3 Homogeneous Experiments of Finite Rank377

4 Approximation by Experiments of Finite Rank387

5 Construction of Distinguished Sequences of Estimates391

CHAPTER 15 Sums of Independent Random Variables399

1 Introduction399

2 Concentration Inequalities401

3 Compactness and Shift-Compactness419

4 Poisson Exponentials and Approximation Theorems423

5 Limit Theorems and Related Results434

6 Sums of Independent Stochastic Processes444

CHAPTER 16 Independent Observations457

1 Introduction457

2 Limiting Distributions for Likelihood Ratios458

3 Conditions for Asymptotic Normality468

4 Tests and Distances475

5 Estimates for Finite Dimensional Parameter Spaces493

6 The Risk of Formal Bayes Procedures509

7 Empirical Measures and Cumulatives529

8 Empirical Measures on Vapnik-？ervonenkis Classes541

CHAPTER 17 Independent Identically Distributed Observations555

1 Introduction555

2 Hilbert Spaces Around a Point556

3 A Special Role for ？ Differentiability in Quadratic Mean573

4 Asymptotic Normality for Rates Other than ？590

5 Existence of Consistent Estimates594

6 Estimates Converging at the ？- Rate604

7 The Behavior of Posterior Distributions614

8 Maximum Likelihood621

9 Some Cases where the Number of Observations Is Random625

Appendix:Results from Classical Analysis634

1 The Language of Set Theory634

2 Topological Spaces638

3 Uniform Spaces640

4 Metric Spaces641

5 Spaces of Functions643

6 Vector Spaces645

7 Vector Lattices650

8 Vector Lattices Arising from Experiments657

9 Lattices of Numerical Functions672

10 Extensions of Positive Linear Functions677

11 Smooth Linear Functionals697

12 Derivatives and Tangents707

Bibliography727

Index737