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Estimating tail decay for stationary sequences via extreme values

Part of: Operations research and management science Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Assaf Zeevi  and
Peter W. Glynn
Assaf Zeevi*
Affiliation:
Columbia University
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Graduate School of Business, Columbia University, 3022 Broadway, New York, NY 10027, USA. Email address: [email protected]
∗∗ Postal address: Management Science and Engineering, Terman Engineering Center, Stanford University, Stanford, CA 94305, USA.
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Abstract

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

Keywords

Extreme valuesestimationtail behaviorstrong mixinguniform mixingminimaxalmost-sure convergence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference 90B20: Traffic problems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 198 - 226
Copyright
Copyright © Applied Probability Trust 2004 

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