Uniform central limit theorems for kernel density estimators

• Published in: Probability theory and related fields Vol. 141; no. 3-4; pp. 333 - 387
• Main Authors: Giné, Evarist, Nickl, Richard
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2008; Springer; Springer Nature B.V
• Subjects: Banach spaces; Central limit theorem; Convergence; Data smoothing; Density; Economics; Estimators; Exact sciences and technology; Finance; General topics; Insurance; Kernels; Limit theorems; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical functions; Mathematical models; Mathematics; Mathematics and Statistics; Nonparametric inference; Operations Research/Decision Theory; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Sciences and techniques of general use; Statistics; Statistics for Business; Studies; Theorems; Theoretical; Smoothed empirical processes; Secondary: 60F05; Kernel density estimation; Uniform central limit theorem; Plug-in property; Primary: 62G07; Density estimation; Empirical process; Probability theory; Bandwith selection; Primary: 62G07; Secondary: 60F05, Kernel density estimation; Kernel estimation; Statistical estimation; Probability distribution; Random vector; Banach space; Kernel estimator; Kernel method; Kernels; Central limit theorem; Convergence rate; Mean error
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-007-0087-9

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.