Tail estimates motivated by extreme value theory. (English) Zbl 0555.62035

Davis, Richard; Resnick, Sidney

Ann. Stat. 12, 1467-1487 (1984).

This paper deals with the problem of estimating the upper tail of a distribution function. The proposed estimator is based on the upper m order statistics from a random sample of size n (m\(\to \infty\), m/n\(\to 0\) as \(n\to \infty)\) and is shown to be consistent for a wide class of distribution functions. Since the empirical mean residual life of the log transformed data and the sample 1-m/n quantile play a key role in the estimator, their joint asymptotic behavior is studied, and rates of convergence of the estimator to the tail are obtained.

Reviewer: J.Melamed

MSC:

62G05	Nonparametric estimation
62G30	Order statistics; empirical distribution functions
62F12	Asymptotic properties of parametric estimators

Keywords:

tail estimation; regular variation; Pareto distribution; order statistics; empirical mean residual life; log transformed data; rates of convergence