Statistics & Probability Letters 2 (1984) 143-146 May 1984 North-Holland A NOTE ON ESTIMATION OF PERCENTILES AND RELIABILITY IN THE EXTREME-VALUE DISTRIBUTION Jerome P. KEATING Division of Mathematics, The University o/Texas at San Antonio, San Antonio, TX 78285, USA Received October 1983 Revised November 1983 Abstract: A method for deriving point estimates of percentiles in the extreme-value distribution conditioned on the ancillary information is given. The resulting conditional median unbiased estimate is the same regardless of the arbitrary choice of equivariant estimators. The conditional median unbiased estimator is also shown to be optimal with respect to Pitman Nearness. Keywords: Weibull failure model, median unbiased estimators, Pitman Nearness. 1. Introduction In a recent expository article, Lawless (1978) illustrates the useful nature of the conditional con- fidence interval approach. The basic concern of Lawless' paper is the interval estimation of param- eters in the extreme-value distribution. The density function of the extreme-value distribution is given by f(x; u, b) = (l/b) exp((x - u)/b) × exp[ - exp((x - u)/b)], (1) -oo <x< q-oo. The primary emphasis of this paper is the estab- lishment of meaningful percentile estimators, which are not sensitive to the choice of an equivariant pair of estimators for (u, b), whereas Lawless strictly concentrated on interval estimates. It is worthwhile to point out that the Weibull distribution can be obtained from (1) using the simple transformation v = exp(x). Thus, the cdf of v is given by F(v;u,b)=l-exp[-(v/exp(u))l/h], v>O. (2) Consequently, the results of this study are also pertinent to problems in which the underlying distribution is Weibull. This fact is especially use- ful since the absence of a pair of sufficient statis- tics in the Weibull and extreme-value distributions has given rise to many point estimators of survival parameters. In the text that follows the proposed estimator of percentiles not only satisfies the property of being invariant to the choice of an equivariant pair but also satisfies an optimal property with respect to Pitman Nearness (PN, see Pitman (1937)). The utility of an estimator which has superior PN to a competing estimator is highlighted by Rao (1980) in his rebuttal of Berkson's (1980) article on the comparison of minimum chi-square and maximum likelihood estimators. Rao (1981) further delin- eates and manifests the importance of this pair- wise measure with the phenomenon that shrinking unbiased estimators to minimum mean squared error estimators does not improve such intrinsic properties as PN. This phenomenon suggests that the best linear invariant estimators (see Mann (1967a, b)) are inferior to best linear unbiased estimators (see Leiblein (1954) and Leiblein and Zelen (1956)) in the sense of PN. Keating and Mason (1983) reinforce Rao's phe- 0167-7152/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland) 143