## The Annals of Statistics

### Refined Pickands estimators of the extreme value index

Holger Drees

#### Abstract

Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index $\beta$ will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative $\beta$ the new estimator is asymptotically superior to previously defined estimators. A simulation study also demonstrates the good small-sample performance. In particular, the estimator proves to be robust against an inappropriate choice of the number of upper order statistics used for estimation.

#### Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2059-2080.

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713647

Digital Object Identifier
doi:10.1214/aos/1034713647

Mathematical Reviews number (MathSciNet)
MR1389865

Zentralblatt MATH identifier
0883.62036

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

#### Citation

Drees, Holger. Refined Pickands estimators of the extreme value index. Ann. Statist. 23 (1995), no. 6, 2059--2080. doi:10.1214/aos/1034713647. https://projecteuclid.org/euclid.aos/1034713647