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

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.