Estimation of marginal excess moments for Weibull-type distributions

Abstract

We consider the estimation of the marginal excess moment (MEM), which is defined for a random vector (X, Y) and a parameter \(\beta >0\) as \(\mathbb {E}[(X-Q_{X}(1-p))_{+}^{\beta }|Y> Q_{Y}(1-p)]\) provided \(\mathbb {E}|X|^{\beta }< \infty \), and where \(y_{+}:=\max (0,y)\), \(Q_{X}\) and \(Q_{Y}\) are the quantile functions of X and Y respectively, and \(p\in (0,1)\). 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 \(p \downarrow 0\) as the sample size \(n \rightarrow \infty \). 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.