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, Volume 24, Issue 4, pp 714–733 | Cite as

Maxima of Gamma random variables and other Weibull-like distributions and the Lambert \(\varvec{W}\) function

  • Armengol Gasull
  • José A. López-Salcedo
  • Frederic UtzetEmail author
Original Paper
  • 191 Downloads

Abstract

In some applied problems of signal processing, the maximum of a sample of \(\chi ^2(m)\) random variables is computed and compared with a threshold to assess certain properties. It is well known that this maximum, conveniently normalized, converges in law to a Gumbel random variable; however, numerical and simulation studies show that the norming constants that are usually suggested are inaccurate for moderate or even large sample sizes. In this paper, we propose, for Gamma laws (in particular, for a \(\chi ^2(m)\) law) and other Weibull-like distributions, other norming constants computed with the asymptotics of the Lambert \(W\) function that significantly improve the accuracy of the approximation to the Gumbel law.

Keywords

Weibull-like distributions Gamma distributions Extreme value theory Lambert function 

Mathematics Subject Classification

60G70 60F05 62G32 41A60 

Notes

Acknowledgments

The first author was partially supported by grants MINECO reference MTM 2013-40998-P and Generalitat de Catalunya reference 2014-SGR568. The second author by the European Space Agency (ESA) through the DINGPOS contract AO/1-5328/06/NL/GLC, and by the Spanish Government and Generalitat de Catalunya through grants TEC2011-28219 and 2014-SGR1586, respectively. The third author by grants MINECO reference MTM2012-33937 and Generalitat de Catalunya reference 2014-SGR422. The authors thank two anonymous referees for their careful reading of the manuscript and their suggestions that helped to improve the paper.

Supplementary material

11749_2015_431_MOESM1_ESM.pdf (511 kb)
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11749_2015_431_MOESM2_ESM.pdf (182 kb)
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Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

  • Armengol Gasull
    • 1
  • José A. López-Salcedo
    • 2
  • Frederic Utzet
    • 1
    Email author
  1. 1.Department of Mathematics, Facultat de CiènciesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Department of Telecommunications and Systems Engineering, Engineering SchoolUniversitat Autònoma de BarcelonaBellaterraSpain

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