Abstract of On The Estimation Of Extreme Tail Probabilities

Research Report SRR95-002

On The Estimation Of Extreme Tail Probabilities

Peter Hall, Ishay Weissmann

Abstract: Applications of extreme value theory to problems of statistical inference typically involve estimating tail probabilities well beyond the range of the data, without the benefit of a concise mathematical model for the sampling distribution. The available model is generally only an asymptotic one. That is, an approximation to probabilities of extreme deviation is supposed, which is assumed to become increasingly accurate as one moves further from the range of the data, but whose concise accuracy is unknown. Quantification of the level of accuracy is essential for optimal estimation of tail probabilities. In the present paper we suggest a practical device, based on a nonstandard application of the bootstrap, for determining empirically the accuracy of the approximation and thereby constructing appropriate estimators. We show that even under simple, classical asymptotic models the problem of calculating an optimal estimator of tail probabilities is unexpectedly complex. It assumes several different forms, depending on the relationship between sample size, n, and the point x at which the tail probability is required. However, our bootstrap method produces first-order optimal estimators of tail probabilities for a very wide range of different relationships between n and x.

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