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Exceedance probability of the integral of a stochastic process

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  • Ferreira, Ana
  • de Haan, Laurens
  • Zhou, Chen

Abstract

Let X={X(s)}s∈S be an almost sure continuous stochastic process (S compact subset of Rd) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.

Suggested Citation

  • Ferreira, Ana & de Haan, Laurens & Zhou, Chen, 2012. "Exceedance probability of the integral of a stochastic process," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 241-257.
    • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:241-257
    DOI: 10.1016/j.jmva.2011.08.020
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    1. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
    2. Lee Fawcett & David Walshaw, 2006. "A hierarchical model for extreme wind speeds," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 55(5), pages 631-646, November.
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    2. Richards, Jordan & Tawn, Jonathan A., 2022. "On the tail behaviour of aggregated random variables," Journal of Multivariate Analysis, Elsevier, vol. 192(C).

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