Estimation of marginal excess moments for Weibull-type distributions

Abstract

AbstractWe consider the estimation of the marginal excess moment (MEM), which is defined for a random vector (X, Y) and a parameter $$\beta >0$$ β > 0 as $$\mathbb {E}[(X-Q_{X}(1-p))_{+}^{\beta }|Y> Q_{Y}(1-p)]$$ E [ ( X - Q X ( 1 - p ) ) + β | Y > Q Y ( 1 - p ) ] provided $$\mathbb {E}|X|^{\beta }< \infty $$ E | X | β < ∞ , 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 ∈ ( 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 ↓ 0 as the sample size $$n \rightarrow \infty $$ 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.