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Extreme values statistics for Markov chains via the (pseudo-) regenerative method

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Abstract

This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in Rootzén (Adv Appl Probab 20:371–390, 1988), the principle underlying our methodology consists of first generating a random number l of approximate pseudo-renewal times τ 1, τ 2, ..., τ l for a sample path X 1, ..., X n drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and then computing submaxima over the approximate cycles thus obtained: \(\max_{1+\tau_1\leq i \leq \tau_2}f(X_i),\;\ldots ,\;\max_{1+\tau_{l-1}\leq i \leq \tau_l}f(X_i)\) for any measurable function f:E→ℝ. Estimators of tail features of the sample maximum max1 ≤ i ≤ n f(X i ) are then constructed by applying standard statistical methods, tailored for the i.i.d. setting, to the submaxima as if they were independent and identically distributed. In particular, the asymptotic properties of extensions of popular inference procedures based on the conditional maximum likelihood theory, such as Hill’s method for the index of regular variation, are thoroughly investigated. Using the same approach, we also consider the problem of estimating the extremal index of the sequence {f(X n )} n ∈ ℕ under suitable assumptions. Eventually, practical issues related to the application of the methodology we propose are discussed and preliminary simulation results are displayed.

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Authors and Affiliations

  1. Université Paris X, MODAL’X Nanterre, Paris, France

    Patrice Bertail

  2. CREST (INSEE), Laboratoire de Statistiques Malakoff, Paris, France

    Patrice Bertail

  3. Telecom Paristech, LTCI - UMR Institut Telecom/CNRS No. 5141, 46, rue Barrault, 75634, Paris Cedex 13, France

    Stéphan Clémençon

  4. INRA - Unité Mét@risk, UR1204, Paris, France

    Jessica Tressou

  5. Department ISOM, The Hong Kong University of Science and Technology, Hong Kong, SAR China

    Jessica Tressou

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  1. Patrice Bertail
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  2. Stéphan Clémençon
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Correspondence to Stéphan Clémençon.

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Bertail, P., Clémençon, S. & Tressou, J. Extreme values statistics for Markov chains via the (pseudo-) regenerative method. Extremes 12, 327–360 (2009). https://doi.org/10.1007/s10687-009-0081-y

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