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Exact extreme value, product, and ratio distributions under non-standard assumptions

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  • Klaus Müller
  • Wolf-Dieter Richter

Abstract

The exact distributions of many functions of random vectors are derived in the literature mainly for the case of a Gaussian vector distribution or under the assumption that the vector follows a spherical or an elliptically contoured distribution. Numerous standard statistical applications are given for these cases. Deriving analogous results, if the sample distribution comes from a large family of probability laws, needs to make use of new analytical tools from the area of exact distribution theory. The present paper provides the application of such tools suitable for deriving the exact cumulative distribution functions and density functions of extreme values, products, and ratios in $$l_{2,p}$$ l 2 , p -symmetrically distributed populations. Accompanying simulation studies are presented in cases of power-exponentially distributed populations and for different sample sizes. As an application, well-known results on the increasing failure rate properties of extremes from Gaussian samples are extended to $$p$$ p -power exponential sample distributions. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Klaus Müller & Wolf-Dieter Richter, 2015. "Exact extreme value, product, and ratio distributions under non-standard assumptions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(1), pages 1-30, January.
    • Handle: RePEc:spr:alstar:v:99:y:2015:i:1:p:1-30
    DOI: 10.1007/s10182-014-0228-2
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    References listed on IDEAS

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    1. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2008. "On the exact distribution of the maximum of absolutely continuous dependent random variables," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 27-35, January.
    2. Jamalizadeh, A. & Balakrishnan, N., 2010. "Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1412-1427, July.
    3. Saralees Nadarajah & Samuel Kotz, 2005. "Linear Combinations, Products and Ratios of t Random Variables," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 89(3), pages 263-280, August.
    4. V. Nekoukhou & M. Alamatsaz, 2012. "A family of skew-symmetric-Laplace distributions," Statistical Papers, Springer, vol. 53(3), pages 685-696, August.
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    Cited by:

    1. Müller K. & Richter W.-D., 2016. "Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-33, February.
    2. Wolf-Dieter Richter, 2017. "Statistical reasoning in dependent p-generalized elliptically contoured distributions and beyond," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-25, December.
    3. Müller K. & Richter W.-D., 2017. "Exact distributions of order statistics from ln,p-symmetric sample distributions," Dependence Modeling, De Gruyter, vol. 5(1), pages 221-245, August.
    4. Klaus Müller & Wolf-Dieter Richter, 2019. "On p-generalized elliptical random processes," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-37, December.
    5. Müller K. & Richter W.-D., 2016. "Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-29, February.

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