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
Author(s): Alexander Korostelev, Olga Korosteleva
Series: Graduate Studies in Mathematics 119
Publisher: American Mathematical Society
Year: 2011
Language: English
Commentary: Includes Errata
Pages: C, X, 246, B
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
