Statistical inference for expectile-based risk measures

TL;DR

This paper studies the statistical properties of expectile-based risk measures, demonstrating their regularity and establishing results like consistency, asymptotic normality, and robustness for related estimators.

Contribution

It provides the first comprehensive analysis of the regularity and asymptotic properties of expectile-based risk measures in both parametric and nonparametric settings.

Findings

Expectile functionals are continuous in the 1-weak topology.

Establishes asymptotic normality and bootstrap consistency.

Demonstrates qualitative robustness of estimators.

Abstract

Expectiles were introduced by Newey and Powell (1987) in the context of linear regression models. Recently, Bellini et al. (2014) revealed that expectiles can also be seen as reasonable law-invariant risk measures. In this article, we show that the corresponding statistical functionals are continuous w.r.t. the $1$-weak topology and suitably functionally differentiable. By means of these regularity results we can derive several properties as consistency, asymptotic normality, bootstrap consistency, and qualitative robustness of the corresponding estimators in nonparametric and parametric statistical models.