An expectile computation cookbook

A substantial body of work in the last 15 years has shown that expectiles constitute an excellent candidate for becoming a standard tool in probabilistic and statistical modeling. Surprisingly, the question of how expectiles may be efficiently calculated has been left largely untouched. We fill this gap by, first, providing a general outlook on the computation of expectiles that relies on the knowledge of analytic expressions of the underlying distribution function and mean residual life function. We distinguish between discrete distributions, for which an exact calculation is always feasible, and continuous distributions, where a Newton-Raphson approximation algorithm can be implemented and a list of exceptional distributions whose expectiles are in analytic form can be given. When the distribution function and/or the mean residual life is difficult to compute, Monte-Carlo algorithms are introduced, based on an exact calculation of sample expectiles and on the use of control variates to improve computational efficiency. We discuss the relevance of our findings to statistical practice and provide numerical evidence of the performance of the considered methods.