Local asymptotic normality in a stationary model for spatial extremes (Q608321)

\textit{L. De Haan} and \textit{T. Pereira}, [Spatial extremes: models for the stationary case. Ann. Stat. 34, No. 1, 146--168 (2006; Zbl 1104.60021)] provided models for spatial extremes in the case of stationarity which depend on just one parameter \(\beta>0\) measuring tail dependence, and proposed different estimators for this parameter. This framework is supplemented by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of \(\beta\). It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.