The Annals of Statistics

On Tail Index Estimation Using Dependent Data

Tailen Hsing

Full-text: Open access


Let $X_1, X_2,\ldots$ be possibly dependent random variables having the same marginal distribution. Consider the situation where $\bar{F}(x) := P\lbrack X_1 > x\rbrack$ is regularly varying at $\infty$ with an unknown index $- \alpha < 0$ which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for $\alpha^{-1}$, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of Hill's estimator extend considerably beyond the independent setting. In addition to some basic results derived under very general conditions, the case where the observations are strictly stationary and satisfy a certain mixing condition is considered in detail. Also a finite moving average sequence is studied to illustrate the results.

Article information

Ann. Statist., Volume 19, Number 3 (1991), 1547-1569.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62G05: Estimation

Order statistics regular variation parameter estimation


Hsing, Tailen. On Tail Index Estimation Using Dependent Data. Ann. Statist. 19 (1991), no. 3, 1547--1569. doi:10.1214/aos/1176348261.

Export citation