Expectiles, Omega Ratios and Stochastic Ordering

Abstract

In this paper we introduce the expectile order, defined by X ≤ _e Y if e _α (X) ≤e _α (Y) for each α ∈ (0, 1), where e _α denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤ _{s t} and ≤ _{c x} , the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤ _{s t} , ≤ _{i c x} and ≤ _e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.