Error bounds for asymptotic expansions of scale mixtures of univariate and multivariate distributions

Let X = [sigma]Z be a scale mixture of a random variable with the scale factor [sigma]. In this paper we consider the expansions G[sigma], k(x) = [Sigma]k - 1j = 0 (j!)-1 a[sigma], j(x) E([sigma][sigma] -1)j as approximations for the distribution function F(x) of X, where [delta] = 1 or -1, k is a positive integer, and a[delta], j(x)'s are defined in terms of the distribution function G(x) of Z. When Z is symmetric, we replace E([sigma][delta] - 1)j by E([sigma]2[delta] - 1)j. The aim of the present paper is to give a unified approach for the error bounds of the two ([delta] = 1, - 1) types of expansion, by expanding the conditional distribution function of X given [sigma], and to extend the results to a scale mixture of a multivariate distribution. We examine in detail the cases when Z is distributed as the gamma distribution and the standard normal distribution.