Tail Estimates Motivated by Extreme Value Theory

An estimate of the upper tail of a distribution function which is based on the upper $m$ order statistics from a sample of size $n(m \rightarrow \infty, m/n \rightarrow 0$ as $n \rightarrow \infty)$ 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.