Nonparametric Functional Estimation

Nonparametric Functional Estimation is a compendium of papers, written by experts, in the area of nonparametric functional estimation. This book attempts to be exhaustive in nature and is written both for specialists in the area as well as for students of statistics taking courses at the postgraduate level. The main emphasis throughout the book is on the discussion of several methods of estimation and on the study of their large sample properties. Chapters are devoted to topics on estimation of density and related functions, the application of density estimation to classification problems, and the different facets of estimation of distribution functions. Statisticians and students of statistics and engineering will find the text very useful.

Preface

List of Notation

1. Estimation of Functional

1.0. Introduction

1.1. Estimability of Functionals

1.2. Necessary and Sufficient Conditions for the Existence of Unbiased Estimators of Functionals

1.3. Asymptotic Efficiency of Estimators of Functionals

Bibliographical Notes

Problems

2. Density Estimation (Univariate Case)

2.0. Introduction

2.1. The Method of Kernels

2.2. The Method of Orthogonal Series

2.3. The Method of Histograms with Fixed Partition

2.4. The Method of Histograms with Random Partition

2.5. The Method of Histosplines

2.6. The Method of Penalty Functions

2.7. The Method of Fourier Inversion

2.8. The Method of Delta Sequences

2.9. A Bayesian Approach

2.10. Comparison of Different Methods

2.11. Optimal Density Estimation

2.12. Invariant Density Estimation

Bibliographical Notes

Problems

3. Density Estimation (Multivariate Case)

3.0. Introduction

3.1. The Method of Kernels

3.2. The Method of Nearest Neighbors

3.3. The Method of Orthogonal Series

3.4. The Method of Delta Sequence

3.5. The Method of Stochastic Approximation

3.6. Further Topics

Bibliographical Notes

Problems

4. Estimation of Functionals Related to Density

4.0. Introduction

4.1. Estimation of Derivatives of a Density

4.2. Estimation of Regression Function

4.3. Estimation of Failure Rate

4.4. Estimation of Functionals of Density and Its Derivatives

4.5. Estimation of Mode

4.6. Further Topics

Bibliographical Notes

Problems

5. Sequential and Recursive Estimation

5.0. Introduction

5.1. Recursive Estimation

5.2. Sequential Estimation

Bibliographical Notes

Problems

6. Estimation for Stochastic Processes

6.1. Discrete Time Stationary Markov Processes

6.2. Discrete Time Stationary φ-Mixing Processes

6.3. Continuous Time Stationary Markov Processes

6.4. Diffusion Processes

Bibliographical Notes

Problems

7. Estimation Under Order Restrictions

7.0. Introduction

7.1. Estimation of Unimodal Density

7.2. Estimation for Distributions with Monotone Failure Rates

7.3. Further Topics

Bibliographical Notes

Problems

8. Nonparametric Discrimination

8.0. Introduction

8.1. Bayes Risk Consistency

8.2. Nearest Neighbor Rules

8.3. Histogram Method for Classification

Bibliographical Notes

Problems

9. Estimation of a Distribution Function

9.0. Introduction

9.1. Estimation by the Method of Rao-Blackwellization

9.2. Estimation by the Method of Kernels

9.3. Estimation by the Method of Stochastic Approximation

9.4. Estimation of a Distribution Function from Incomplete Observations (Univariate Case)

9.5. Estimation in the Competing Risks Problem

9.6. Estimation of a Distribution Function from Incomplete Observations (Bivariate Case)

Bibliographical Notes

Problems

10. Estimation of a Mixing Distribution

10.0. Introduction

10.1. Estimation for Finite Mixtures

10.2. Estimation for Arbitrary Mixtures

Bibliographical Notes

Problems

11. Bayes Estimation

11.0. Introduction

11.1. Estimation of a Distribution Function

11.2. Estimation of a Distribution Function from Incomplete Data

11.3. Estimation of a Symmetric Distribution Function

11.4. Estimation of a Functional of a Distribution Function

Bibliographical Notes

Problems

References

Author Index

Subject Index