Nonparametric confidence intervals for the integral of a function of an unknown density

Abstract

Given a random sample of size n from an unknown distribution function F on ℝ with finite derivatives and density f, we wish to estimate for a smooth function L. Examples are∈t f 2, the differential entropy and the Kullback–Leibler distance. We estimate f using a kernel estimate [fcirc] based on a kernel of order p, say. We show that {[fcirc](x i ), i=1, …, s} satisfies the Cornish–Fisher assumption with respect to m=nh. It follows that the corresponding estimate θˆ has a bias of magnitude O(h q +m −1), where p≤q≤2p depends on L. We show that the variance of θˆ has magnitude O(n −1) for a suitable bandwidth. For the regular case, we give one-sided and two-sided confidence intervals for θ with errors of magnitude O(M −1/2) and O(M −1), where M=nh 2. We present simulation studies to show the practical values of the results.