Skip to main content
SpringerLink
Log in

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form sδ/gr - 1, where ρ(> 0) and δ(= ±1) are parameters indicating the types of expansion. For δ = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis (δ = -1) or by numerical computation (δ = 1).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Fujikoshi, Y. (1987). Error bounds for asymptotic expansions of scale mixtures of distributions, Hiroshima Math. J., 17, 309–324.

    Google Scholar 

  • Fujikoshi, Y. (1993). Error bounds for asymptotic approximations of some distribution functions, Multivariate Analysis: Future Directions (ed. C. R. Rao), 181–208, North-Holland, Amsterdam.

    Google Scholar 

  • Fujikoshi, Y. and Shimizu, R. (1989). Asymptotic expansions of some mixtures of univariate and multivariate distributions, J. Multivariate Anal., 30, 279–291.

    Google Scholar 

  • Fujikoshi, Y. and Shimizu, R. (1990). Asymptotic expansions of some distributions and their error bounds—the distributions of sums of independent random variables and scale mixtures, Sugaku Expositions, 3, 75–96.

    Google Scholar 

  • Hall, P. (1979). On measures of the distance of a mixture from its parent distribution, Stochastic Process. Appl., 8, 357–365.

    Google Scholar 

  • Heyde, C. C. and Leslie, J. R. (1976). On moment measures of departure from the normal and exponential laws, Stochastic Process. Appl., 4, 317–328.

    Google Scholar 

  • Shimizu, R. (1987). Error bounds for asymptotic expansion of the scale mixtures of the normal distribution, Ann. Inst. Statist. Math., 30, 611–622.

    Google Scholar 

  • Shimizu, R. (1989). Expansion of the scale mixtures of the gamma distribution, J. Statist. Plann. Inference, 21, 305–314.

    Google Scholar 

  • Shimizu, R. (1995). Expansion of the scale mixtures of the multivariate normal distribution with error bound evaluated in L1-norm, J. Multivariate Anal., 53, 126–138.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo, 106, Japan

    Ryoichi Shimizu

  2. Department of Mathematics, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima, 739, Japan

    Yasunori Fujikoshi

Authors
  1. Ryoichi Shimizu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasunori Fujikoshi
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Shimizu, R., Fujikoshi, Y. Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures. Annals of the Institute of Statistical Mathematics 49, 285–297 (1997). https://doi.org/10.1023/A:1003162929574

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1003162929574

Search

Navigation