The Annals of Applied Probability

Tail index estimation for dependent data

Sidney Resnick and Catalin Stărică

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Abstract

A popular estimator of the index of regular variation in heavy-tailed models is Hill's estimator. We discuss the consistency of Hill's estimator when it is applied to certain classes of heavy-tailed stationary processes. One class of processes discussed consists of processes which can be appropriately approximated by sequences of m-dependent random variables and special cases of our results show the consistency of Hill's estimator for (i) infinite moving averages with heavy-tail innovations, (ii) a simple stationary bilinear model driven by heavy-tail noise variables and (iii) solutions of stochastic difference equations of the form $$Y_t = A_tY_{t-1} + Z_t, -\infty < t < \infty$$ where ${(A_n, Z_n), \quad -\infty < n < \infty}$ are iid and the Z's have a regularly varying tail probabilities. Another class of problems where our methods work successfully are solutions of stochastic difference equations such as the ARCH process where the process cannot be successfully approximated by m-dependent random variables. A final class of models where Hill estimator consistency is proven by our tail empirical process methods is the class of hidden semi-Markov models.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 4 (1998), 1156-1183.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903376

Digital Object Identifier
doi:10.1214/aoap/1028903376

Mathematical Reviews number (MathSciNet)
MR1661160

Zentralblatt MATH identifier
0942.60037

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G10: Stationary processes

Keywords
Hill estimator tail estimator heavy tails tail empirical process ARCH model bilinear model moving average hidden Markov model

Citation

Resnick, Sidney; Stărică, Catalin. Tail index estimation for dependent data. Ann. Appl. Probab. 8 (1998), no. 4, 1156--1183. doi:10.1214/aoap/1028903376. https://projecteuclid.org/euclid.aoap/1028903376


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