An estimate of the upper tail of a distribution function which is based on the upper m order statistics from a sample of size n(m → ∞, m/n → 0 as n → ∞) is shown to be consistent for a wide class of distribution functions. The empirical mean residual life of the log transformed data and the sample $1 - m/n$ quantile play a key role in the estimate. The joint asymptotic behavior of the empirical mean residual life and sample $1 - m/n$ quantile is determined and rates of convergence of the estimate to the tail are derived.
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Davis et al. (1984) studied this question.