#### Title

On moment inequalities of the supremum of empirical processes with applications to kernel estimation

#### Abbreviated Journal Title

Stat. Probab. Lett.

#### Keywords

empirical process; moment inequalities; moment generating functions; upper bounds; kernel density estimates; uniform consistency; Statistics & Probability

#### Abstract

Let X-1,...,X-n be a random sample from a distribution function F. Let F-n(x) = (1/n) Sigma(i=1)(n)I(X-i less than or equal to x) denote the corresponding empirical distribution function. The empirical process is defined by D-n(x) = rootn/F-n(x)-F(x)\. In this note, upper bounds are found for E(D-n) and for E(e(tDn)), where D-n = sup(x)D(n)(x). An extension to the two sample case is indicated. As one application, upper bounds are obtained for E(W-n), where, W-n = sup(x)\(f) over cap (n)(x) - f(x)\, with (f) over cap (n)(x) = (1/nh) Sigma(i=1)(n) k((x - X-i)/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. (C) 2002 Elsevier Science B.V. All rights reserved.

#### Journal Title

Statistics & Probability Letters

#### Volume

57

#### Issue/Number

3

#### Publication Date

1-1-2002

#### Document Type

Article

#### Language

English

#### First Page

215

#### Last Page

220

#### WOS Identifier

#### ISSN

0167-7152

#### Recommended Citation

"On moment inequalities of the supremum of empirical processes with applications to kernel estimation" (2002). *Faculty Bibliography 2000s*. 3034.

http://stars.library.ucf.edu/facultybib2000/3034

## Comments

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