A Weissman-type estimator of the conditional marginal expected shortfall
Résumé
The marginal expected shortfall is an important risk measure in finance, which has been extended recently in the case where the random variables of main interest (Y^{(1)}, Y^{(2)}) are observed together with a covariate X \in R^d. This leads to the concept of conditional marginal expected shortfall. It is defined as \theta_p(x_0)=E[Y^{(1)} | Y^{(2)} \geq Q_{Y^{(2)}}(1-p|x_0); x_0], where p is small and Q_{Y^{(2)}} denotes the quantile function of Y^{(2)}. In this paper, we propose an estimator for \theta_p(x_0) allowing extrapolation outside the Y^{(2)} - data range, i.e., valid for p < 1/n. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory.
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