Accurately approximating extreme value statistics

Abstract

We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to \infty$, 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.

arXiv comment17 pages, 10 figures

References in corpus (17)

• The large deviation approach to statistical mechanics
• Statistical Models of Fracture
• Universality Classes for Extreme Value Statistics
• Extreme value statistics of correlated random variables: a pedagogical review
• The Single Big Jump Principle in Physical Modelling
• Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
• Distribution of extreme first passage times of diffusion
• Finite-size scaling in extreme statistics
• Renormalization group theory for finite-size scaling in extreme statistics
• Large deviations of the maximum of independent and identically distributed random variables
• Exact extreme value statistics at mixed order transitions
• Transport in disordered systems: the single big jump approach
• Primordial Non-Gaussianity and Extreme-Value Statistics of Galaxy Clusters
• Statistics of the longest interval in renewal processes
• Renormalization flow in extreme value statistics
• Extreme value theory for constrained physical systems
• Condensation and extremes for a fluctuating number of independent random variables

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