Non-uniform error bounds for asymptotic expansions of scale mixtures of distributions

Let X = [sigma]Z be the scale mixture of Z with the scale factor [sigma] > 0. We consider two type expansions G[delta],k(x) and [Phi][delta],k(x) as the approximations to the distribution function F(x) of X. In this paper we derive non-uniform error bounds in approximating F(x) by the asymptotic expansions G[delta],k(x) and [Phi][delta],k(x). The non-uniform bounds are improvements on the uniform bounds in the tail part of the distribution. The results are applied to the asymptotic expansions of t- and F-distributions.