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

Author

Listed:
  • Armengol Gasull
  • José López-Salcedo
  • Frederic Utzet

Abstract

In some applied problems of signal processing, the maximum of a sample of $$\chi ^2(m)$$ χ 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 $$\chi ^2(m)$$ χ 2 ( m ) law) and other Weibull-like distributions, other norming constants computed with the asymptotics of the Lambert $$W$$ W function that significantly improve the accuracy of the approximation to the Gumbel law. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • Armengol Gasull & José López-Salcedo & Frederic Utzet, 2015. "Maxima of Gamma random variables and other Weibull-like distributions and the Lambert $$\varvec{W}$$ W function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 714-733, December.
    • Handle: RePEc:spr:testjl:v:24:y:2015:i:4:p:714-733
    DOI: 10.1007/s11749-015-0431-9
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    Cited by:

    1. Ji, Lanpeng & Peng, Xiaofan, 2023. "Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 418-452.
    2. Brice Ozenne & Esben Budtz-Jørgensen & Sebastian Elgaard Ebert, 2023. "Controlling the familywise error rate when performing multiple comparisons in a linear latent variable model," Computational Statistics, Springer, vol. 38(1), pages 1-23, March.

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