AStA Advances in Statistical Analysis

, Volume 99, Issue 1, pp 1–30 | Cite as

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

Original Paper

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 \(l_{2,p}\)-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 \(p\)-power exponential sample distributions.

Keywords

Geometric measure representation \(l_{2 , p}\)-generalized arc-length measure Intersection-percentage function \(p\)-power exponential distribution Lifetime analysis IFR property Simulation 

Notes

Acknowledgments

The authors are grateful to the Associated Editor and a reviewer for giving valuable hints which led to improvements in the presentation of the material.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of RostockRostockGermany

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