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November 1998 Tail index estimation for dependent data
Sidney Resnick, Catalin Stărică
Ann. Appl. Probab. 8(4): 1156-1183 (November 1998). DOI: 10.1214/aoap/1028903376

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.

Citation

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Sidney Resnick. Catalin Stărică. "Tail index estimation for dependent data." Ann. Appl. Probab. 8 (4) 1156 - 1183, November 1998. https://doi.org/10.1214/aoap/1028903376

Information

Published: November 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0942.60037
MathSciNet: MR1661160
Digital Object Identifier: 10.1214/aoap/1028903376

Subjects:
Primary: 60G70
Secondary: 60G10

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.8 • No. 4 • November 1998
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