Model-free inference on extreme dependence via waiting times

Abstract

A variety of methods have been proposed for inference about extreme dependence for multivariate or spatially-indexed stochastic processes and time series. Most of these proceed by first transforming data to some specific extreme value marginal distribution, often the unit Fréchet, then fitting a family of max-stable processes to the transformed data and exploring dependence within the framework of that model. The marginal transformation, model selection, and model fitting are all possible sources of misspecification in this approach. We propose an alternative model-free approach, based on the idea that substantial information on the strength of tail dependence and its temporal structure are encoded in the distribution of the waiting times between exceedances of high thresholds at different locations. We propose quantifying the strength of extremal dependence and assessing uncertainty by using statistics based on these waiting times. The method does not rely on any specific underlying model for the process, nor on asymptotic distribution theory. The method is illustrated by applications to climatological, financial, and electrophysiology data. To put the proposed approach within the context of the existing literature, we construct a class of spacetime-indexed stochastic processes whose waiting time distributions are available in closed form by endowing the support points in de Haan's spectral representation of max-stable processes with random birth times, velocities, and lifetimes, and applying Smith's model to these processes. We show that waiting times in this model are stochatically decreasing in mean speed, and the sample mean of the waiting times obeys a central limit theorem with a uniform convergence rate under mild conditions. This indicates that our procedure can be implemented in this setting using standard $t$ statistics and associated hypothesis tests.