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Two-Moment Approximations for Maxima

33 State Road, 3rd Floor, Suite F, Princeton, New Jersey 08540
Operations Research Center, MIT, Cambridge, Massachusetts 02139
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
chuck.crow{at}ccrow.net
dag3141{at}mit.edu
ww2040{at}columbia.edu

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.

Subject classifications: probability; distributions; maximum of independent random variables; probability; distributions; two-moment approximations.
History: Received June 2003; revision received July 2006; accepted July 2006.