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Estimation of spatial max-stable models using threshold exceedances. (English) Zbl 1325.62105
Summary: Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block maxima, typically annual, is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. We focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by A. W. Ledford and J. A. Tawn [Biometrika 83, No. 1, 169–187 (1996; Zbl 0865.62040)] when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension [H. Rootzén and N. Tajvidi, Bernoulli 12, No. 5, 917–930 (2006; Zbl 1134.62028)] of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study where both processes in a domain of attraction of a max-stable process and max-stable processes are successively considered as time replications, according to different degrees of spatial dependency. Results put forward how the nature of the time replications influences the bias of estimations and highlight the choice of each approach regarding to the strength of the spatial dependencies and the threshold choice.

MSC:
62G32 Statistics of extreme values; tail inference
60G52 Stable stochastic processes
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