Mathematical statistics. Asymptotic minimax theory

Preface
Part 1  Parametric Models
     Chapter 1  The Fisher Efficiency
          1.1. Statistical Experiment
          1.2. The Fisher Information
          1.3. The Cramer-Rao Lower Bound
          1.4. Efficiency of Estimators
          Exercises
     Chapter 2  The Bayes andMinimax Estimators
          2.1. Pitfalls of the Fisher Efficiency
          2.2. The Bayes Estimator
          2.3. Minimax Estimator. Connection Between Estimators
          2.4. Limit of the Bayes Estimator and Minimaxity
          Exercises
     Chapter 3  Asymptotic Minimaxity
          3.1. The Hodges Example
          3.2. Asymptotic Minimax Lower Bound
          3.3. Sharp Lower Bound. Normal Observations
          3.4. Local Asymptotic Normality (LAN)
          3.5. The Hellinger Distance
          3.6. Maximum Likelihood Estimator
          3.7. Proofs of Technical Lemmas
          Exercises
     Chapter 4  Some Irregular Statistical Experiments
          4.1. Irregular Models: Two Examples
          4.2. Criterion for Existence of the Fisher Information
          4.3. Asymptotically Exponential Statistical Experiment
          4.4. Minimax Rate of Convergence
          4.5. Sharp Lower Bound
          Exercises
     Chapter 5  Change-Point Problem
          5.1. Model of Normal Observations
          5.2. Maximum Likelihood Estimator of Change Point
          5.3. Minimax Limiting Constant
          5.4. Model of Non-Gaussian Observations
          5.5. Proofs of Lemmas
          Exercises
     Chapter 6  Sequential Estimators
          6.1. The Markov Stopping Time
          6.2. Change-Point Problem. Rate of Detection
          6.3. Minimax Limit in the Detection Problem.
          6.4. Sequential Estimation in the Autoregressive Model
               6.4.1. Heuristic Remarks on MLE
               6.4.2. On-Line Estimator
          Exercises
     Chapter 7  Linear Parametric Regression
          7.1. Definitions and Notations
          7.2. Least-Squares Estimator
          7.3. Properties of the Least-Squares Estimator
          7.4. Asymptotic Analysis of the Least-Squares Estimator
               7.4.1. Regular Deterministic Design
               7.4.2. Regular Random Design
          Exercises
Part 2  Nonparametric Regression
     Chapter 8  Estimation in Nonparametric Regression
          8.1. Setup and Notations
          8.2. Asymptotically Minimax Rate of Convergence. Definition
          8.3. Linear Estimator
               8.3.1. Definition
               8.3.2. The Nadaraya-Watson Kernel Estimator
          8.4. Smoothing Kernel Estimator
          Exercises
     Chapter 9  Local Polynomial Approximation  of the Regression Function
          9.1. Preliminary Results and Definition
          9.2. Polynomial Approximation and Regularity of Design
               9.2.1. Regular Deterministic Design
               9.2.2. Random Uniform Design
          9.3. Asymptotically Minimax Lower Bound
               9.3.1. Regular Deterministic Design
          9.4. Proofs of Auxiliary Results
          Exercises
     Chapter 10  Estimation of Regression in Global Norms
          10.1. Regressogram
          10.2. Integral L2-Norm Risk for the Regressogram
          10.3. Estimation in the Sup-Norm
          10.4. Projection on Span-Space and Discrete MISE
          10.5. Orthogonal Series Regression Estimator
               10.5.1. Preliminaries
               10.5.2. Discrete Fourier Series and Regression
          Exercises
     Chapter 11  Estimation by Splines
          11.1. In Search of Smooth Approximation
          11.2. Standard B-splines
          11.3. Shifted B-splines and Power Splines
          11.4. Estimation of Regression by Splines
          11.5. Proofs of Technical Lemmas
          Exercises
     Chapter 12  Asymptotic Optimality in Global Norms
          12.1. Lower Bound in the Sup-Norm
          12.2. Bound in £ 2-Norm. Assouad\'s Lemma
          12.3. General Lower Bound
          12.4. Examples and Extensions
          Exercises
Part 3  Estimation in Nonparametric Models
     Chapter 13  Estimation of Functionals
          13.1. Linear Integral Functionals
          13.2. Non-Linear Functionals
          Exercises
     Chapter 14  Dimension and Structure in Non parametric Regression
          14.1. Multiple Regression Model
          14.2. Additive regression
          14.3. Single-Index Model
               14.3.1. Definition
               14.3.2. Estimation of Angle
               14.3.3. Estimation of Regression Function
          14.4. Proofs of Technical Results
          Exercises
     Chapter 15  Adaptive Estimation
          15.1. Adaptive Rate at a Point. Lower Bound
          15.2. Adaptive Estimator in the Sup-Norm
          15.3. Adaptation in the Sequence Space
          15.4. Proofs of Lemmas
          Exercises
     Chapter 16  Testing of Nonparametric Hypotheses
          16.1. Basic Definitions
               16.1.1. Parametric Case.
               16.1.2. Nonparametric Case
          16.2. Separation Rate in the Sup-Norm
          16.3. Sequence Space. Separation Rate in the L2-Norm
          Exercises
Bibliography
Index of Notation
Index
List of  Errata
Back Cover