Two-Moment Approximations for Maxima

Charles S. Crow (), David Goldberg () and Ward Whitt ()
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Charles S. Crow: 33 State Road, 3rd Floor, Suite F, Princeton, New Jersey 08540
David Goldberg: Operations Research Center, MIT, Cambridge, Massachusetts 02139
Ward Whitt: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027

Operations Research, 2007, vol. 55, issue 3, 532-548

Abstract: 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.

Keywords: probability; distributions; maximum of independent random variables; probability; distributions; two-moment approximations (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:55:y:2007:i:3:p:532-548

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