Interpreting Expectiles

Abstract

Expectiles are the most popular generalized quantile, but they can be enigmatic to unfamiliar users. We organize nine interpretations for expectiles along different perspectives. An expectile is the minimizer of an asymmetric least squares criterion, making it a weighted average but also meaning that the expectile is the true mean of the distribution in two special cases. Specifically, an expectile of a distribution is a value that would be the mean if values above it were more likely to occur than they actually are. Expectiles summarize a distribution in a manner similar to quantiles, but also quantiles are expectiles in location models and expectiles are quantiles, albeit not always of the original distribution. Expectiles are also m-estimators, m-quantiles, and Lp-quantiles, families containing the majority of simple statistics commonly in use.