Local asymptotic normality in (δ-neighborhoods of standard generalized Pareto processes

De Haan and Pereira (2006) provided models for spatial extremes in the case of stationarity, which depend on just one parameter beta > 0 measuring tail dependence, and they proposed different estimators for this parameter. This framework was supplemented by Falk (2011) by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold, yielding in particular asymptotic efficient estimators. The estimators investigated in these papers are based on a finite set of points t(1),...,t(d), at which observations are taken. We generalize this approach in the context of functional extreme value theory (EVT). This more general framework allows estimation over some spatial parameter space, i.e., the finite set of points t(1),...,t(d) is replaced by t is an element of [a,b]. In particular, we derive efficient estimators of beta based on those processes in a sample of iid processes in C[0,1] which exceed a given threshold function. (C) 2011 Elsevier B.V. All rights reserved.