Title

On Moment Inequalities Of The Supremum Of Empirical Processes With Applications To Kernel Estimation

Keywords

Empirical process; Kernel density estimates; Moment generating functions; Moment inequalities; Uniform consistency; Upper bounds

Abstract

Let X1,...,Xn be a random sample from a distribution function F. Let Fn(x) = (1/n) ∑i=1n I(Xi ≤ x) denote the corresponding empirical distribution function. The empirical process is defined by Dn(x) = √n Fn(x) - F(x). In this note, upper bounds are found for E(Dn) and for E(etDn), where Dn = supx Dn(x). An extension to the two sample case is indicated. As one application, upper bounds are obtained for E(Wn), where, Wn = supx f̂n(x) - f(x), with f̂n(x) = (1/nh) ∑i=1n k((x - Xi)/h) is the celebrated "kernel" density estimate of f(x), the density corresponding to F(x) and an optimal bandwidth is selected based on Wn. Analogous results for the kernel estimate of F are also mentioned. © 2002 Elsevier Science B.V. All rights reserved.

Publication Date

4-15-2002

Publication Title

Statistics and Probability Letters

Volume

57

Issue

3

Number of Pages

215-220

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/S0167-7152(02)00029-9

Socpus ID

0037089156 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0037089156

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