Acknowledgments
To the Reader
l General Introduction
2 Summary of Contents and Historical Notes
CHAPTER l
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 Z
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-Cervonenkis Classes
CHAPTER 17
Independent Identically Distributcd 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
