Edgeworth and Cornish Fisher expansions and confidence intervals for the distribution, density and

Abstract

We show that kernel density estimates of bandwidth h=h(n)→0 satisfy the Cornish-Fisher assumption with parameter m=nh. This allows Cornish-Fisher expansions about the normal for standardized and Studentized kernel density estimates. The expansions given are formal and the conditions for existence/validity are not explored. The expansions lead to first order confidence intervals (CIs) of level 1−ω +O(n−β), where β =p/(2p+ 2) for one-sided CIs and β = p/(p+1) for two-sided CIs, where p is the order of the kernel used. The second order one- and two-sided CIs are given with β =2p/(2p+3) and β =2p/(p+2). We show how to choose the bandwidth for asymptotic optimality.