The Annals of Statistics

Tail Estimates Motivated by Extreme Value Theory

Richard Davis and Sidney Resnick

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Abstract

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.

Article information

Source
Ann. Statist., Volume 12, Number 4 (1984), 1467-1487.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346804

Digital Object Identifier
doi:10.1214/aos/1176346804

Mathematical Reviews number (MathSciNet)
MR760700

Zentralblatt MATH identifier
0555.62035

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions 62F12: Asymptotic properties of estimators

Keywords
Tail estimation regular variation Pareto distributions mean residual life intermediate order statistics

Citation

Davis, Richard; Resnick, Sidney. Tail Estimates Motivated by Extreme Value Theory. Ann. Statist. 12 (1984), no. 4, 1467--1487. doi:10.1214/aos/1176346804. https://projecteuclid.org/euclid.aos/1176346804


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