Kernel estimation of extreme regression risk measures

Abstract : The Regression Conditional Tail Moment (RCTM) is the risk measure defined as the moment of order $b ≥ 0$ of a loss distribution above the upper α-quantile where $α ∈ (0, 1$) and when a covariate information is available. The purpose of this work is first to establish the asymptotic properties of the RCTM in case of extreme losses, i.e when $α → 0$ is no longer fixed, under general extreme-value conditions on their distribution tail. In particular, no assumption is made on the sign of the associated extreme-value index. Second, the asymptotic normality of a kernel estimator of the RCTM is established, which allows to derive similar results for estimators of related risk measures such as the Regression Conditional Tail Expectation/Variance/Skewness. When the distribution tail is upper bounded, an application to frontier estimation is also proposed. The results are illustrated both on simulated data and on a real dataset in the field of nuclear reactors reliability.
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Jonathan El Methni, Laurent Gardes, Stephane Girard. Kernel estimation of extreme regression risk measures. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.359--398. ⟨10.1214/18-EJS1392⟩. ⟨hal-01393519v3⟩

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