Accurately approximating extreme value statistics

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Abstract

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N → ∞, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with N is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.

Original languageEnglish
Article number315205
JournalJournal of Physics A: Mathematical and Theoretical
Volume54
Issue number31
DOIs
StatePublished - Aug 2021

Bibliographical note

Publisher Copyright:
© 2021 IOP Publishing Ltd.

Funding

The support of the Israel Science Foundation, Grant No. 1898/17, is acknowledged.

FundersFunder number
Israel Science Foundation1898/17

    Keywords

    • Lambert scaling
    • extreme value theory
    • large deviations theory
    • limiting distribution
    • slow convergence

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