Kernel Smoothing

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.

Preface Introduction Introduction Density estimation and histograms About this book Options for reading this book Bibliographical notes Univariate kernel density estimation Introduction 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 Exercises Bandwidth selection Introduction 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 Exercises Multivariate kernel density estimation Introduction The multivariate kernel density estimator Asymptotic MISE approximations Exact MISE calculations Choice of multivariate kernel Choice of smoothing parametrisation Bandwidth selection Bibliographical notes Exercises Kernel regression Introduction 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 Exercises Selected extra topics Introduction 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 Exercises Appendixes 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 References Index

M.P. Wand, M.C. Jones