## The Annals of Statistics

### On Asymptotic Normality of Hill's Estimator for the Exponent of Regular Variation

#### Abstract

It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number $k_n$ of extreme order statistics used to construct it tends to infinity appropriately with the sample size $n.$ As our main result, we derive a general condition which can be used to determine the optimal $k_n$ explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition is simplified under appropriate assumptions and then applied to several examples.

#### Article information

Source
Ann. Statist., Volume 13, Number 2 (1985), 743-756.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176349551

Digital Object Identifier
doi:10.1214/aos/1176349551

Mathematical Reviews number (MathSciNet)
MR790569

Zentralblatt MATH identifier
0606.62019

JSTOR