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L1-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling's generalized

Author

Listed:
  • Fujikoshi, Y.
  • Ulyanov, V.V.
  • Shimizu, R.

Abstract

This paper is concerned with the distribution of a multivariate scale mixture variate X=(X1,...,Xp)' with Xi=SiZi, where Z1,...,Zp are i.i.d. random variables, Si>0(i=1,...,p), and {S1,...,Sp} is independent of {Z1,...,Zp}. First we obtain L1-norm error bounds for an asymptotic expansion of the density function of X in the multivariate case as well as in the univariate case. Then the results are applied in obtaining error bounds for asymptotic expansions of the null distribution of Hotelling's generalized -statistic. The special features of our results are that our error bounds are given in explicit and computable forms. Further, their orders are the same as ones of the usual order estimates, and hence the paper provides a new proof for validity of the asymptotic expansions.

Suggested Citation

  • Fujikoshi, Y. & Ulyanov, V.V. & Shimizu, R., 2005. "L1-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling's generalized," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 1-19, September.
    • Handle: RePEc:eee:jmvana:v:96:y:2005:i:1:p:1-19
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    References listed on IDEAS

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    1. Hall, Peter, 1979. "On measures of the distance of a mixture from its parent distribution," Stochastic Processes and their Applications, Elsevier, vol. 8(3), pages 357-365, May.
    2. Ryoichi Shimizu & Yasunori Fujikoshi, 1997. "Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(2), pages 285-297, June.
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    Cited by:

    1. Gerd Christoph & Vladimir V. Ulyanov, 2021. "Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes," Mathematics, MDPI, vol. 9(7), pages 1-28, April.
    2. 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.
    3. Fujikoshi, Yasunori & Ulyanov, Vladimir V., 2006. "Error bounds for asymptotic expansions of Wilks' lambda distribution," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1941-1957, October.
    4. Gerd Christoph & Vladimir V. Ulyanov, 2023. "Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples," Mathematics, MDPI, vol. 11(8), pages 1-18, April.

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