Modelling extreme values of processes observed at irregular time step. Application to significant wave height.
Résumé
The distribution of extremes such as flood peaks, maximum wave height or minimum daily returns over annual or other time intervals is of common interest to many disciplines including the natural and social sciences. This work is motivated by the analysis of extreme values from times series of significant wave heights observed in North Atlantic. One of these time series exhibits missing data (buoy data) and another one irregular time sampling (satellite data). This situation is frequent when considering environmental data sets and new statistical methods are needed to analyze the extremal behavior of such time series. The method proposed in this work consists in assuming that the behavior of the process above a high threshold is well approximated by a max-stable process which parameters are estimated by maximizing a composite likelihood function. The consistency of these estimates is established. Then, using an extensive set of simulated time series, we assess the finite-sample behavior of our estimates for small to medium sample sizes and compare them to other available estimation methods proposed in the literature for analyzing the extremal behavior of stochastic processes on the basis of standard validation statistics. Finally, a detailed study of significant wave height data is performed. It is shown that the proposed methodology may be used to estimate characteristics of extreme significant wave height at any location in the ocean from altimeter satellite data.
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