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.

Title: Modelling dependence between hydrological and meteorological variables measured on several stations Author: Bc. Marie Turčičová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Daniela Jarušková CSc., Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mathematics Abstract: The aim of the thesis is to explore the dependence of daily discharge averages of the Opava river on high daily precipitation values in its basin. Three methods are presented that can be used for analyzing the dependence between high values of random variables. Their application on the studied data is also given. First it is the tail-dependence coefficient that measures the dependence between high values of two continuous random variables. The model for the high quantiles of the discharge at a given precipitation value was first determined non-parametrically by quantile regression and then parametrically through the peaks-over-threshold (POT) method. Keywords: extremal dependence, tail-dependence coefficient, quantile regression, peaks over threshold method