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. - 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. - 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.