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

Christopher Withers and Saralees Nadarajah

Journal of Nonparametric Statistics, 2011, vol. 23, issue 4, 943-966

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 f2, 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](xi), 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(hq+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=nh2. We present simulation studies to show the practical values of the results.

Date: 2011
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DOI: 10.1080/10485252.2011.576762

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