A class of distortion measures generated from expectile and its estimation

We construct a specific form of piecewise distortion function which can distort a random risk to its expectile. After analyzing this kind of distortion functions, we define a class of distortion functions which are generated from random variables. The consistent estimation of the expectile distortion parameter is given by the maximum empirical likelihood method. The expectile distortion not only inherits the good properties of concave distortion measures but also has its own advantages. Since that, we discuss the potential usage of this measure and imagine a new premium principle based on the non self form of this measure.