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Consistency of Hill's estimator for dependent data

Part of: Parametric inference Stochastic processes Limit theorems Distribution theory Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Sidney Resnick  and
Cătălin Stărică
Sidney Resnick*
Affiliation:
Cornell University
Cătălin Stărică*
Affiliation:
Cornell University
*
Postal address of both authors: School of ORIE, Cornell University Ithaca, NY 14853, USA.
Postal address of both authors: School of ORIE, Cornell University Ithaca, NY 14853, USA.
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Abstract

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

Keywords

REGULAR VARIATIONTIME SERIESTAIL EMPIRICAL PROCESSHEAVY TAILSINFINITEORDER MOVING AVERAGESAUTOREGRESSIVE PROCESSES

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 60G35: Signal detection and filtering 60G57: Random measures 62E17: Approximations to distributions (nonasymptotic) 62F10: Point estimation 62G30: Order statistics; empirical distribution functions
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 139 - 167
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

∗

Partially supported by NSF Grant DMS-9100027 at Cornell University. Some support was also received from NSA Grant 92G-116.

∗∗

Supported by NSF Grant DMS-9100027 at Cornell University.

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