Asymptotic Properties of the Empirical Spatial Extremogram

TL;DR

This paper extends the extremogram to spatial data, establishing its asymptotic properties and demonstrating its practical utility in analyzing spatial extremal dependence, especially in environmental data like rainfall.

Contribution

It introduces the spatial extremogram, proves a central limit theorem for its empirical version, and applies it to real rainfall data, expanding extremogram analysis to spatial contexts.

Findings

Central limit theorem established for the empirical spatial extremogram.

Applicable to max-moving average and Brown-Resnick processes.

Demonstrated effectiveness through simulation and rainfall data analysis.

Abstract

The extremogram, proposed by Davis and Mikosch (2008), is a useful tool for measuring extremal dependence and checking model adequacy in a time series. We define the extremogram in the spatial domain when the data is observed on a lattice or at locations distributed as a Poisson point process in d-dimensional space. Under mixing and other conditions, we establish a central limit theorem for the empirical spatial extremogram. We show these conditions are applicable for max-moving average processes and Brown-Resnick processes and illustrate the empirical extremogram's performance via simulation. We also demonstrate its practical use with a data set related to rainfall in a region in Florida.