1st Edition

Kernel Smoothing

By M.P. Wand, M.C. Jones Copyright 1995
    226 Pages
    by Chapman & Hall

    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.




    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




    M.P. Wand, M.C. Jones