CHAPTER 1 Experiments—Decision Spaces

1 Introduction

2 Vector Lattices—L-Spaces—Transitions

3 Experiments—Decision Procedures

4 A Basic Density Theorem

5 Building Experiments from Other Ones

6 Representations—Markov Kernels

CHAPTER 2 Some Results from Decision Theory:Deficiencies

1 Introduction

2 Characterization of the Spaces of Risk Functions:Minimax Theorem

3 Deficiencies;Distances

4 The Form of Bayes Risks—Choquet Lattices

CHAPTER 3 Likelihood Ratios and Conical Measures

1 Introduction

2 Homogeneous Functions of Measures

3 Deficiencies for Binary Experiments:Isometries

4 Weak Convergence of Experiments

5 Boundedly Complete Experiments

6 Convolutions:Hellinger Transforms

7 The Blackwell-Sherman-Stein Theorem

CHAPTER 4 Some Basic Inequalities

1 Introduction

2 Hellinger Distances:L1-Norm

3 Approximation Properties for Likelihood Ratios

4 Inequalities for Conditional Distributions

CHAPTER 5 Sufficiency and Insufficiency

1 Introduction

2 Projections and Conditional Expectations

3 Equivalent Definitions for Sufficiency

4 Insufficiency

5 Estimating Conditional Distributions

CHAPTER 6 Domination,Compactness,Contiguity

1 Introduction

2 Definitions and Elementary Relations

3 Contiguity

4 Strong Compactness and a Result of D.Lindae

CHAPTER 7 Some Limit Theorems

1 Introduction

2 Convergence in Distribution or in Probability

3 Distinguished Sequences of Statistics

4 Lower-Semicontinuity for Spaces of Risk Functions

5 A Result on Asymptotic Admissibility

CHAPTER 8 Invariance Properties

1 Introduction

2 The Markov-Kakutani Fixed Point Theorem

3 A Lifting Theorem and Some Applications

4 Automatic Invariance of Limits

5 Invariant Exponential Families

6 The Hunt-Stein Theorem and Related Results

CHAPTER 9 Infinitely Divisible,Gaussian,and Poisson Experiments

1 Introduction

2 Infinite Divisibility

3 Gaussian Experiments

4 Poisson Experiments

5 A Central Limit Theorem

CHAPTER 10 Asymptotically Gaussian Experiments:Local Theory

1 Introduction

2 Convergence to a Gaussian Shift Experiment

3 A Framework which Arises in Many Applications

4 Weak Convergence of Distributions

5 An Application of a Martingale Limit Theorem

6 Asymptotic Admissibility and Minimaxity

CHAPTER 11 Asymptotic Normality—Global

1 Introduction

2 Preliminary Explanations

3 Construction of Centering Variables

4 Definitions Relative to Quadratic Approximations

5 Asymptotic Properties of the Centerings ？

6 The Asymptotically Gaussian Case

7 Some Particular Cases

8 Reduction to the Gaussian Case by Small Distortions

9 The Standard Tests and Confidence Sets

10 Minimum x2 and Relatives

CHAPTER 12 Posterior Distributions and Bayes Solutions

1 Introduction

2 Inequalities on Conditional Distributions

3 Asymptotic Behavior of Bayes Procedures

4 Approximately Gaussian Posterior Distributions

CHAPTER 13 An Approximation Theorem for Certain Sequential Experiments

1 Introduction

2 Notations and Assumptions

3 Basic Auxiliary Lemmas

4 Reduction Theorems

5 Remarks on Possible Applications

CHAPTER 14 Approximation by Exponential Families

1 Introduction

2 A Lemma on Approximate Sufficiency

3 Homogeneous Experiments of Finite Rank

4 Approximation by Experiments of Finite Rank

5 Construction of Distinguished Sequences of Estimates

CHAPTER 15 Sums of Independent Random Variables

1 Introduction

2 Concentration Inequalities

3 Compactness and Shift-Compactness

4 Poisson Exponentials and Approximation Theorems

5 Limit Theorems and Related Results

6 Sums of Independent Stochastic Processes

CHAPTER 16 Independent Observations

1 Introduction

2 Limiting Distributions for Likelihood Ratios

3 Conditions for Asymptotic Normality

4 Tests and Distances

5 Estimates for Finite Dimensional Parameter Spaces

6 The Risk of Formal Bayes Procedures

7 Empirical Measures and Cumulatives

8 Empirical Measures on Vapnik-？ervonenkis Classes

CHAPTER 17 Independent Identically Distributed Observations

1 Introduction

2 Hilbert Spaces Around a Point

3 A Special Role for ？ Differentiability in Quadratic Mean

4 Asymptotic Normality for Rates Other than ？

5 Existence of Consistent Estimates

6 Estimates Converging at the ？- Rate

7 The Behavior of Posterior Distributions

8 Maximum Likelihood

9 Some Cases where the Number of Observations Is Random

Appendix:Results from Classical Analysis

1 The Language of Set Theory

2 Topological Spaces

3 Uniform Spaces

4 Metric Spaces

5 Spaces of Functions

6 Vector Spaces

7 Vector Lattices

8 Vector Lattices Arising from Experiments

9 Lattices of Numerical Functions

10 Extensions of Positive Linear Functions

11 Smooth Linear Functionals

12 Derivatives and Tangents

Bibliography

Index