Dependent conditional tail expectation for extreme levels

Abstract

We consider the estimation of the dependent conditional tail expectation, defined for a random vector (X,Y) with X≥0 as E(X|X>QX(1−p),Y>QY(1−p)), when E(X)<∞, and where QX and QY denote the quantile functions of X and Y, respectively. The distribution of X is assumed to be of Pareto-type while the distribution of Y is kept general. Using extreme-value arguments we introduce an estimator for this risk measure for the situation p≤1/n, where n is the number of available observations, i.e., focus is on estimation with extrapolation. The convergence in distribution of our estimator is established and its finite sample performance is illustrated on a simulation study. The method is then applied on wind gusts data set.