Expectile Asymptotics

TL;DR

This paper investigates the asymptotic behavior of sample expectiles, establishing conditions for their consistency and normality, and exploring their distributional limits under various distributional assumptions.

Contribution

It provides a comprehensive analysis of the asymptotic distribution of expectiles, including uniform consistency, asymptotic normality, and stable laws for heavy-tailed distributions.

Findings

Expectiles are uniformly consistent under finite mean.

Asymptotic normality requires continuity at the expectile for finite second moments.

Heavy-tailed distributions lead to stable law limits for expectiles.

Abstract

We discuss in detail the asymptotic distribution of sample expectiles. First, we show uniform consistency under the assumption of a finite mean. In case of a finite second moment, we show that for expectiles other then the mean, only the additional assumption of continuity of the distribution function at the expectile implies asymptotic normality, otherwise, the limit is non-normal. For a continuous distribution function we show the uniform central limit theorem for the expectile process. If, in contrast, the distribution is heavy-tailed, and contained in the domain of attraction of a stable law with $1 < \alpha < 2$, then we show that the expectile is also asymptotically stable distributed. Our findings are illustrated in a simulation section.