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 \(\beta >0\) $\beta >0$ as \(\mathbb {E}[(X-Q_{X}(1-p))_{+}^{\beta }|Y> Q_{Y}(1-p)]\) $\mathbb{E}[{(X-Q_X(1-p))}_{+}^{\beta }|Y>Q_Y(1-p)]$ provided \(\mathbb {E}|X|^{\beta }< \infty \) ${\mathbb{E}|X|}^{\beta }<\infty$ , and where \(y_{+}:=\max (0,y)\) $y_{+}:=max(0,y)$ , \(Q_{X}\) $Q_X$ and \(Q_{Y}\) $Q_Y$ are the quantile functions of X and Y respectively, and \(p\in (0,1)\) $p\in (0,1)$ . 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 \(p \downarrow 0\) $p\downarrow 0$ as the sample size \(n \rightarrow \infty \) $n\to \infty$ . 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.