Estimation of marginal excess moments for Weibull-type distributions

We consider the estimation of the marginal excess moment (MEM), which is defined for a random vector (X, Y) and a parameter β>0 as E[(X-QX(1-p))+β|Y>QY(1-p)] provided E|X|β<∞, and where y+:=max(0,y), QX and QY are the quantile functions of X and Y respectively, and p∈(0,1). Our interest is in the situation where the random variable X is of Weibull-type while the distribution of Y is kept general, the extreme dependence structure of (X, Y) converges to that of a bivariate extreme value distribution, and we let p↓0 as the sample size n→∞. By using extreme value arguments we introduce an estimator for the marginal excess moment and we derive its limiting distribution. The finite sample properties of the proposed estimator are evaluated with a simulation study and the practical applicability is illustrated on a dataset of wave heights and wind speeds.