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, c 2 , 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 c 2 (ge) 1, convolutions of exponentials for c 2 (le) 1, and gamma for all c 2 . 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 (ge) n * required for the asymptotic extreme-value approximations to be effective. The threshold n * tends to increase as c 2 increases above one or decreases below one.