Local Polynomial Modelling and Its Applications

Transcription

1 Local Polynomial Modelling and Its Applications J. Fan Department of Statistics University of North Carolina Chapel Hill, USA and I. Gijbels Institute of Statistics Catholic University oflouvain Louvain-la-Neuve, Belgium CHAPMAN & HALL London Weinheim New York Tokyo Melbourne Madras

2 Contents Preface xiii 1 Introduction Prom linear regression to nonlinear regression Local modelling Bandwidth selection and model complexity Scope of the book Implementation of nonparametric techniques Further reading 12 2 Overview of existing methods Introduction Kernel estimators Nadaraya-Watson estimator Gasser-Miiller estimator Limitations of a local constant fit Local polynomial fitting and derivative estimation Local polynomial fitting Derivative estimation Locally weighted scatter plot smoothing Robust locally weighted regression An example Wavelet thresholding, Orthogonal series based methods Basic ingredient of multiresolution analysis Wavelet shrinkage estimator Discrete wavelet transform Spline smoothing Polynomial spline Smoothing spline 43

3 CONTENTS 2.7 Density estimation Kernel density estimation Regression view of density estimation Wavelet estimators Logspline method Bibliographic notes 55 Framework for local polynomial regression Introduction Advantages of local polynomial fitting Bias and variance Equivalent kernels Ideal choice of bandwidth Design adaptation property Automatic boundary carpentry Universal optimal weighting scheme Which order of polynomial fit to use? Increases of variability It is an odd world Variable order approximation Best linear smoothers Best linear smoother at interior: optimal rates and constants Best linear smoother at boundary Minimax efficiency of local polynomial fitting Modulus of continuity Best rates and nearly best constant Fast computing algorithms Binning implementation Updating algorithm Complements Bibliographic notes 105 Automatic determination of model complexity Introduction Rule of thumb for bandwidth selection Estimated bias and variance Confidence intervals Residual squares criterion Residual squares criterion , Constant bandwidth selection Variable bandwidth selection 122

4 CONTENTS ix Computation and related issues Refined bandwidth selection Improving rates of convergence Constant bandwidth selection Variable bandwidth selection Variable bandwidth and spatial adaptation Qualification of spatial adaptation Comparison with wavelets Smoothing techniques in use Example 1: modelling and model diagnostics Example 2: comparing two treatments Example 3: analyzing a longitudinal data set A blueprint for local modelling Other existing methods Normal reference method Cross-validation Nearest neighbor bandwidth Plug-in ideas Sheather and Jones' bandwidth selector Complements Bibliographic notes Applications of local polynomial modelling Introduction Censored regression Preliminaries Censoring unbiased transformation Local polynomial regression An asymptotic result Proportional hazards model Partial likelihood Local partial likelihood Determining model complexity Complete likelihood Generalized linear models Exponential family models Quasi-likelihood and deviance residuals Local quasi-likelihood Bias and variance Bandwidth selection Robust regression Robust methods 199

5 x CONTENTS Quantile regression Simultaneous estimation of location and scale functions Complements Bibliographic notes Applications in nonlinear time series Introduction Nonlinear prediction Mixing conditions Local polynomial fitting Estimation of conditional densities Percentile and expectile regression Regression percentile Expectile regression Spectral density estimation Smoothed log-periodogram Maximum local likelihood method Smoothed periodogram Sensitivity measures and nonlinear prediction Sensitivity measures Noise amplification Nonlinear prediction error Complements Bibliographic notes Local polynomial regression for multivariate data Introduction Generalized additive models Generalized partially linear single-index models Partially linear models Single-index models Generalized partially linear single-index models Modelling interactions Interactions in generalized additive models Interactions in generalized partially linear single-index models Multivariate adaptive regression splines Sliced inverse regression Local polynomial regression as a building block Robustness Local linear regression in the multivariate setting 297

6 CONTENTS Multivariate local linear regression estimator Bias and variance Optimal weight function Efficiency 7.9 BiblioEraDhic notes XI References 307 Author index 330 Subject index 336