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

Let X1,...,Xn be a random sample from a distribution function F. Let Fn(x)=(1/n)[summation operator]i=1nI(Xi[less-than-or-equals, slant]x) denote the corresponding empirical distribution function. The empirical process is defined by 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, with 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.