Asymptotic Methods in Statistical Decision Theory

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.- 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.- 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.- 4 Some Basic Inequalities.- 1 Introduction.- 2 Hellinger Distances: L1-Norm.- 3 Approximation Properties for Likelihood Ratios.- 4 Inequalities for Conditional Distributions.- 5 Sufficiency and Insufficiency.- 1 Introduction.- 2 Projections and Conditional Expectations.- 3 Equivalent Definitions for Sufficiency.- 4 Insufficiency.- 5 Estimating Conditional Distributions.- 6 Domination, Compactness, Contiguity.- 1 Introduction.- 2 Definitions and Elementary Relations.- 3 Contiguity.- 4 Strong Compactness and a Result of D. Lindae.- 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.- 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.- 9 Infinitely Divisible,Gaussian, and Poisson Experiments.- 1 Introduction.- 2 Infinite Divisibility.- 3 Gaussian Experiments.- 4 Poisson Experiments.- 5 A Central Limit Theorem.- 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.- 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 $$\hat{Z}$$.- 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 ?2 and Relatives.- 12 Posterior Distributions and Bayes Solutions.- 1 Introduction.- 2 Inequalities on Conditional Distributions.- 3 Asymptotic behavior of Bayes Procedures.- 4 Approximately Gaussian Posterior Distributions.- 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.- 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.- 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.- 16Independent 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.- 17 Independent Identically Distributed Observations.- 1 Introduction.- 2 Hilbert Spaces Around a Point.- 3 A Special Role for $$\sqrt{n}$$: Differentiability in Quadratic Mean.- 4 Asymptotic Normality for Rates Other than $$\sqrt{n}$$.- 5 Existence of Consistent Estimates.- 6 Estimates Converging at the $$\sqrt{n}$$-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.