Mathematical Statistics: Asymptotic Minimax Theory (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 119)

This book is designed to bridge the gap between traditional textbooks in statistics and more advanced books that include the sophisticated nonparametric techniques. It covers topics in parametric and nonparametric large-sample estimation theory. The exposition is based on a collection of relatively simple statistical models. It gives a thorough mathematical analysis for each of them with all the rigorous proofs and explanations. The book also includes a number of helpful exercises. Prerequisites for the book include senior undergraduate/beginning graduate-level courses in probability and statistics. Readership: Graduate students and research mathematicians interested in mathematical statistics 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 Chapter 1. The Fisher Efficiency Chapter 2. The Bayes And Minimax Estimators Chapter 3. Asymptotic Minimaxity Chapter 4. Some Irregular Statistical Experiments Chapter 5. Change-point Problem Chapter 6. Sequential Estimators Chapter 7. Linear Parametric Regression Chapter 8. Estimation In Nonparametric Regression Chapter 9. Local Polynomial Approximation Of Regression Function Chapter 10. Estimation Of Regression In Global Norms Chapter 11. Estimation By Splines Chapter 12. Asymptotic Optimality In Global Norms Chapter 13. Estimation Of Functionals Chapter 14. Dimension And Structure In Nonparametric Regression Chapter 15. Adaptive Estimation Chapter 16. Testing Of Nonparametric Hypotheses Alexander Korostelev, Olga Korosteleva. Includes Bibliographical References (p. 239) And Indexes. "This book is designed to bridge the gap between traditional textbooks in statistics and more advanced books that include the sophisticated nonparametric techniques. It covers topics in parametric and nonparametric large-sample estimation theory. The exposition is based on a collection of relatively simple statistical models. It gives a thorough mathematical analysis for each of them with all the rigorous proofs and explanations. The book also includes a number of helpful exercises."--Publisher's description