Extremile Regression

Résumé

Extremiles are a least squares alternative to quantiles, determined by probability-weighted moments rather than tail probabilities. They benefit from several interpretations and closed form expressions that are equivalent for continuous distributions, and they characterize a distribution just as quantiles do. Their regression versions similarly define a least squares analog of regression quantiles. We give a comprehensive overview of the state of the art regarding probabilistic and statistical properties of unconditional extremiles and their regression counterparts and provide a comparison between extremiles and other important classes of indicators for the description of unconditional and conditional distributions on real data examples.