Effects of smoothing on distribution approximations

Abstract

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We show that a number of apparently disparate problems, involving distribution approximations in the presence of discontinuities, are actually closely related. One class of such problems involves developing bootstrap approximations to the distribution of a sample mean when the sample includes both ordinal and continuous data. Another class involves smoothing a lattice distribution so as to overcome rounding errors in the normal approximation. A third includes kernel methods for smoothing distribution estimates when constructing confidence bands. Each problem in these classes may be modelled in terms of sampling from a mixture of a continuous and a lattice distribution. We quantify the proportion of the continuous component that is sufficient to smooth away difficulties caused by the lattice part. The proportion is surprisingly small ? it is only a little larger than n_1 logn, where ? denotes sample size. Therefore, very few continuous variables are required in order to render a continuity correction unnecessary. The implications of this result in the problem of sampling both ordinal and continuous data are discussed, and numerical aspects are described through a simulation study. The result is also used to characterise bandwidths that are appropriate for smoothing distribution estimators in the confidence band problem. In this setting an empirical method for bandwidth choice is suggested, and a particularly simple derivation of Edgeworth expansions is given.