Conditional Expectation as Quantile Derivative
نویسنده
چکیده
For a linear combination ∑ uj Xj of random variables, we are interested in the partial derivatives of its α-quantile Qα(u) regarded as a function of the weight vector u = (uj). It turns out that under suitable conditions on the joint distribution of (Xj) the derivatives exist and coincide with the conditional expectations of the Xi given that ∑ uj Xj takes the value Qα(u). Moreover, using this result, we deduce formulas for the derivatives with respect to the ui for the so-called expected shortfall E [ ∣∣∑uj Xj −Qα(u)∣∣δ ∣∣ ∑uj Xj ≤ Qα(u) ], with δ ≥ 1 fixed. Finally, we study in some more detail the coherence properties of the expected shortfall in case δ = 1.
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