Statistical Experiments and Decisions

The following sections are included:

• Contents

• Preface

The following sections are included:

• Statistical Experiment (Main Definitions, the Problem of Comparison of Experiments)

• Randomization of Statistical Experiments. I. Markov Kernels. Stochastic Operators

• Examples of Comparison of Experiments with the Help of Randomization

• The Main Components of the General Decision Theory

• Randomization of Statistical Experiments. II. General case

• Comparison of Experiments via Decision Problems

• Deficiency δ(ɛ,ɛ*) and Δ-Distance Δ(ɛ,ɛ*)

• On Calculation of Deficiency δ(ɛ, ɛ*) and Δ-Distance with the Help of Bayes Risks

• Some Explicit Formulas for Deficiency and Δ-Distance

• A Particular Case of Comparison of Experiments (Due to Blackwell, Shannon, Fisher)

• The Hellinger and Mellin Transformations

• Absolutely Continuous and Contiguous Probability Measures

The following sections are included:

• Strong (Δ−) Convergence

• Weak (w−) Convergence

• Reasons for Introducing the λ-Convergence (Sufficient Statistics, Localization, Reparametrization, LAN Property)

• λ-Convergence (Definition, Examples)

• λ-Convergence and Accompanying Experiments

• Uniform Versions of λ-Convergence

• λ(K)-Convergence and Asymptotic Minimax Theorems

• Comparison of Various Kinds of Convergence of Statistical Experiments

• Convergence of Statistical Decisions and Estimators

• Proof of Lemma about “Reconstruction”

• Differentiable Experiments and Stochastic Expansions for Likelihood

• Extended λ-Convergence

• Contiguity of Statistical Experiments. I

• Contiguity of Statistical Experiments. II

The following sections are included:

• Definition of (γ, Γ)-Models. Examples

• Generalized Bayes Approach for (γ, Γ)-Models. Lemma About Averaging

• Lower Bound of Minimax Risk for (γ, Γ)-Models. Minimax Estimators

• Structure of Regular Estimators for Some (γ, Γ)-Models

• λ-Convergence of Statistical Experiments to (γ, Γ)-Models. Asymptotically Minimax Estimation

• Approximation of Statistical Experiments by (γ, Γ; φ)-Models. I. Discrete Time

• Approximation of Statistical Experiments by (γ, Γ; φ)-Models. II. Continuous Time

The following sections are included:

• Local λ-Convergence. Main Definitions

• Asymptotic Minimax Theorems under Local λ-Convergence

• Global Estimation. Preliminary Considerations

• Connectedness Equation for Local λ-Convergence

• Asymptotic Efficiency of Global Estimators under Local λ-Convergence. General Case

• Connectedness Equation for Transitive and Weakly Transitive Limit Experiments

• Global Estimation under Local λ-Convergence to Transitive and Weakly Transitive Experiments

The following sections are included:

• Parameter Estimation

• Convergence of Statistical Experiments for First Order Autoregressive Models
  • Gaussian errors
  • Non-Gaussian errors: stable autoregression
  • Non-Gaussian errors: explosive autoregression

• Asymptotic Efficient Minimax Estimation of Autoregression Parameter
  • Gaussian case. Lower bounds
  • Gaussian case. Upper bounds
  • Non-Gaussian case

• Sequential Estimation

The following sections are included:

• Bibliography

• Index

• List of Symbols

• List of Conditions