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Journal Articles Electronic Journal of Statistics Year : 2018

## Kernel estimation of extreme regression risk measures

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Jonathan El Methni
Laurent Gardes
• Function : Author
• PersonId : 911294
Stéphane Girard

#### 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.

#### Domains

Mathematics [math] Statistics [math.ST]

### Dates and versions

hal-01393519 , version 1 (07-11-2016)
hal-01393519 , version 2 (15-03-2017)
hal-01393519 , version 3 (14-11-2017)

### Identifiers

• HAL Id : hal-01393519 , version 3
• DOI :

### Cite

Jonathan El Methni, Laurent Gardes, Stéphane Girard. Kernel estimation of extreme regression risk measures. Electronic Journal of Statistics , 2018, 12 (1), pp.359--398. ⟨10.1214/18-EJS1392⟩. ⟨hal-01393519v3⟩

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