Author | Cam, Lucien Le. author |
---|---|
Title | Asymptotic Methods in Statistical Decision Theory [electronic resource] / by Lucien Le Cam |
Imprint | New York, NY : Springer New York, 1986 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4946-7 |
Descript | XXVI, 742 p. online resource |
1 ExperimentsโDecision Spaces -- 1 Introduction -- 2 Vector LatticesโL-SpacesโTransitions -- 3 ExperimentsโDecision Procedures -- 4 A Basic Density Theorem -- 5 Building Experiments from Other Ones -- 6 RepresentationsโMarkov 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 RisksโChoquet 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 MarkovโKakutani 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 NormalityโGlobal -- 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