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

    Authors

    I. A. Ahmad

    Comments

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    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

    WOS:000176787300001

    ISSN

    0167-7152

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