The Annals of Applied Probability

Tail index estimation for dependent data

Sidney Resnick and Catalin Stărică

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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

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

First available in Project Euclid: 9 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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