On asymptotic normality of Hill’s estimator for the exponent of regular variation. (English) Zbl 0606.62019

Haeusler, E.; Teugels, Jozef L.

Ann. Stat. 13, 743-756 (1985).

It is shown that Hill’s estimator [B. M. Hill, ibid. 3, 1163-1174 (1975; Zbl 0323.62033)] 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.

MSC:

62E20	Asymptotic distribution theory in statistics
62G05	Nonparametric estimation
62G30	Order statistics; empirical distribution functions

Keywords:

limit theorems; Hill’s estimator; exponent of regular variation; asymptotically normal; extreme order statistics; regularly varying upper tail

Citations:

Zbl 0323.62033