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Name: Marine Demangeot

Talk Title: Estimation of the extremal coefficient function based on a single observation

Abstract: The extremal coefficient function is a bivariate measure of spatial dependence for stationary max-stable processes [1]. It is usually estimated from time series, when the spatial object under study is observed through time (e.g. extreme precipitations, extreme temperatures, high concentrations of pollution in the air). However, in some cases, such types of data cannot be accessed: only one or just a few records are made available. This is the case, for instance, in mining resources estimation, soil contamination evaluation or any other applications where the phenomenon of interest either varies too slowly across time to hope for a decent time series, or is too expensive to sample from. This situation is rarely addressed in the spatial extremes community, contrary to Geostatistics, which typically deals with such issues. A basic geostatistical tool is the so-called variogram [2], which is also a bivariate measure of spatial dependence. Considering the indicator variogram, above some threshold, of a stationary max-stable random field, we propose a new nonparametric estimator of the extremal coefficient function based on the variogram's Nadaraya-Watson estimator. The latter has been studied by [3] and [4]; from their work, we derive asymptotic properties of our estimator when it is computed from a single spatial set of observations. Namely, under some assumptions, we show that it is consistent and asymptotically normal. These results are illustrated by numerical experiments and a comparison with the well-known F-madogram based estimator [5] is performed. An application on a real dataset is also presented.

[1] M. Schlather and J.A. Tawn, A dependence measure for multivariate and spatial extreme values: Properties and inference, Biometrika, 2003

[2] J.-P. Chilès and P. Delfiner, Geostatistics: Modeling Spatial Uncertainty, Wiley series in probability and statistics, second edition, 2012

[3] P.H. Garcia-Soidan, M. Febrero-Bande and W. Gonzalez-Manteiga, Nonparametric kernel estimation of an isotropic variogram, Journal of Statistical Planning and Inference, 2004

[4] P.H. Garcia-Soidan, Asymptotic normality of the Nadaraya-Watson semivariogram estimators, TEST, 2007

[5] D. Cooley, P. Naveau and P. Poncet Variograms for spatial max-stable random fields, pages 370-390, Springer New-York, 2006

This talk is a contributed talk at EVA 2021.