## The Annals of Statistics

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

### On Tail Index Estimation Using Dependent Data

#### 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

**Keywords**

Order statistics regular variation parameter 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