Kernel Smoothing  book cover
1st Edition

Kernel Smoothing

ISBN 9780412552700
Published December 1, 1994 by Chapman and Hall/CRC
224 Pages

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Book Description

Kernel smoothing refers to a general methodology for recovery of underlying structure in data sets.The basic principle is that local averaging or smoothing is performed with respect to a kernel function.

This book provides uninitiated readers with a feeling for the principles, applications, and analysis of kernel smoothers. This is facilitated by the authors' focus on the simplest settings, namely density estimation and nonparametric regression. They pay particular attention to the problem of choosing the smoothing parameter of a kernel smoother, and also treat the multivariate case in detail.

Kernel Smoothing is self-contained and assumes only a basic knowledge of statistics, calculus, and matrix algebra. It is an invaluable introduction to the main ideas of kernel estimation for students and researchers from other discipline and provides a comprehensive reference for those familiar with the topic.

More information on the book, and the accompanying R package can be found here.

Table of Contents




Density estimation and histograms

About this book

Options for reading this book

Bibliographical notes

Univariate kernel density estimation


The univariate kernel density estimator

The MSE and MISE criteria

Order and asymptotic notation; Taylor expansion

Order and asymptotic notation

Taylor expansion

Asymptotic MSE and MISE approximations

Exact MISE calculations

Canonical kernels and optimal kernel theory

Higher-older kernels

Measuring how difficult a density is to estimate

Modifications of the kernel density estimations

Local kernel density estimators

Variable kernel density estimators

Transformation kernel density estimators

Density estimation at boundaries

Density derivative estimation

Bibliographical notes


Bandwidth selection


Quick and simple bandwidth selectors

Normal scale rules

Oversmoothed bandwidth selection rules

Least squares cross-validation

Biased cross-validation

Estimation of density functionals

Plug-in bandwidth selection

Direct plug in rules

Solve-the-equation rules

Smoothed cross-validation bandwidth selection

Comparison of bandwidth selection

Theoretical performance

Practical advice

Bibliographical notes


Multivariate kernel density estimation


The multivariate kernel density estimator

Asymptotic MISE approximations

Exact MISE calculations

Choice of multivariate kernel

Choice of smoothing parametrisation

Bandwidth selection

Bibliographical notes


Kernel regression


Local polynomial kernel estimators

Asymptotic MSE approximations: linear case

Fixed equally spaced design

Random design

Asymptotic MSE approximations: general case

Behaviour near the boundary

Comparison with other kernel estimators

Asymptotic comparison

Effective kernels

Derivative estimation

Bandwidth selection

Multivariate nonparametric regression

Bibliographical notes


Selected extra topics


Kernel density estimation in other settings

Dependent data

Length biased data

Right-censored data

Data measured with error

Hazard function estimation

Spectral density estimation

Likelihood-based regression models

Intensity function estimation

Bibliographical notes



A Notation

B Tables

C Facts about normal densities

C.1 Univariate normal densities

C.2 Multivariate normal densities

C.3 Bibliographical notes

D Computation of kernel estimators

D.1 Introduction

D.2 The binned kernel density estimator

D.3 Computation of kernel functional estimates

D.4 Computation of kernel regression estimates

D.5 Extension to multivariate kernel smoothing

D.6 Computing practicalities

D.7 Bibliographical notes



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