A survey and comparison of methods for estimating extreme right tail-area quantiles

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Abstract
The purpose of this paper is to survey many of the methods for estimating extreme right tail-area quantiles in order to determine which method or methods gives the best approximations. The problem is to find a good estimate of xp defined by 1 - F(x p) = p where p is a very small number for a random sample from an unknown distribution. An extension of this problem is to determine the number of largest order statistics that should be used to make an estimate. From extensive computer simulations trying to minimize relative error, conclusions can be drawn based on the value of p. For p = .02, the exponential tail method by Breiman, et al using a method by Pickands for determining the number of order statistics to use works best for light to heavy tailed distributions. For extremely heavy tailed distributions, a method proposed by Hosking and Wallis seems to be the most accurate at p = .02 and p = .002. The quadratic tail method by Breiman, et al appears best for light to moderately heavy tailed distributions at p = .002 and for all distributions at p = .0002.