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.