Two-Moment Approximations for Maxima

Charles S. Crow, David Goldberg, Ward Whitt
2007 Operations Research  
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 ,
more » ... on (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 1, convolutions of exponentials for c 2 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 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. Subject classifications: probability, distributions: maximum of independent random variables; probability, distributions: two-moment approximations. Area of review: Stochastic Models.
doi:10.1287/opre.1060.0375 fatcat:max27ywymbfz7hvby6ilwn5z6a