## The Annals of Statistics

### On Tail Index Estimation Using Dependent Data

Tailen Hsing

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348261

Digital Object Identifier
doi:10.1214/aos/1176348261

Mathematical Reviews number (MathSciNet)
MR1126337

Zentralblatt MATH identifier
0738.62026

JSTOR
links.jstor.org

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

#### Citation

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