Generalized madogram and pairwise dependence of maxima over two disjoint regions of a random field

Quantifying dependence between extreme events occurring at several locations of a random field is a fundamental issue in applied spatial extreme value analysis. For max-stable processes, an important measure of pairwise dependence is the $\lambda$-madogram. In this paper we propose a generalization of this coefficient that enables the analysis of dependence between maxima over two disjoint regions of locations and present some of its properties. Two estimatores of this function are given. Finally, we define a multivariate maxima of moving maxima random field, compute its generalized madogram for some choices of regions of locations and analyze the performance of the proposed estimators through a simulation study.