Extremal Covariance Matrices

Abstract

The tail dependence coefficient (TDC) is a natural tool to describe extremal dependence. Estimation of the tail dependence coefficient can be performed via empirical process theory. In case of extremal independence, the limit degenerates and hence one cannot construct a test for extremal independence. In order to deal with this issue, we consider an analog of the covariance matrix, namely the extremogram matrix, whose entries depend only on extremal observations. We show that under the null hypothesis of extremal independence and for finite dimension d ≥ 2, the largest eigenvalue of the sample extremogram matrix converges to the maximum of d independent normal random variables. This allows us to conduct an hypothesis testing for extremal independence by means of the asymptotic distribution of the largest eigenvalue. Simulation studies are performed to further illustrate this approach.