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.
Highlights
That is, regression on a parameter that generalizes the mean and characterizes the tail behaviour of a distribution, has been introduced by Newey and Powell (1987) as an alternative to more standard quantile regression; Breckling and Chambers (1988) considered regression based on more general asymmetric M-estimators
In this note we study in detail the statistical, that is, asymptotic properties of the sample expectiles
Somewhat surprisingly and in contrast to the mean, for τ = 1/2 we find that even under the assumption of a finite second moment, the sample expectile is only asymptotically normal if the distribution function F is continuous at μτ (F ), otherwise, the limit distribution is non-normal
Summary
That is, regression on a parameter that generalizes the mean and characterizes the tail behaviour of a distribution, has been introduced by Newey and Powell (1987) as an alternative to more standard quantile regression; Breckling and Chambers (1988) considered regression based on more general asymmetric M-estimators. Somewhat surprisingly and in contrast to the mean, for τ = 1/2 we find that even under the assumption of a finite second moment, the sample expectile is only asymptotically normal if the distribution function F is continuous at μτ (F ), otherwise, the limit distribution is non-normal. If F has a jump at μτ (F ), we show in Section 2.3 that under the assumption of a finite second moment, the asymptotic distribution of the sample expectile is non-normal. Using a non-standard version of the functional delta-method allows them to treat both the case of dependent data as well as expectiles of parametric estimates of the distribution They only consider the case of a finite second moment (they even assume slightly more) and a distribution which is continuous at the expectiles, and further do not investigate properties of the expectile process
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