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

Author

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  • Fujikoshi, Yasunori
  • Shimizu, Ryoichi

Abstract

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.

Suggested Citation

  • Fujikoshi, Yasunori & Shimizu, Ryoichi, 1989. "Error bounds for asymptotic expansions of scale mixtures of univariate and multivariate distributions," Journal of Multivariate Analysis, Elsevier, vol. 30(2), pages 279-291, August.
    • Handle: RePEc:eee:jmvana:v:30:y:1989:i:2:p:279-291
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    Cited by:

    1. Siotani, Minoru & Wakaki, Hirofumi, 2006. "Contributions to multivariate analysis by Professor Yasunori Fujikoshi," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1914-1926, October.

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