Two-Moment Approximations for Maxima

We introduce and investigate approximations for the probability distribution of the maximum of n independent and identically distributed nonnegative random variables, in terms of the number n and the first few moments of the underlying probability distribution, assuming the distribution is unbounded above but does not have a heavy tail. Because the mean of the underlying distribution can immediately be factored out, we focus on the effect of the squared coefficient of variation (SCV, c2, variance divided by the square of the mean). Our starting point is the classical extreme-value theory for representative distributions with the given SCV---mixtures of exponentials for c2 ≥ 1, convolutions of exponentials for c2 ≤ 1, and gamma for all c2. We develop approximations for the asymptotic parameters and evaluate their performance. We show that there is a minimum threshold n*, depending on the underlying distribution, with n ≥ n* required for the asymptotic extreme-value approximations to be effective. The threshold n* tends to increase as c2 increases above one or decreases below one.

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