# Expectiles, Omega Ratios and Stochastic Ordering

19 Pages Posted: 8 Jun 2016

See all articles by Fabio Bellini

## Fabio Bellini

University of Milano Bicocca - Dipartimento di Statistica e Metodi Quantitativi

## Bernhard Klar

Karlsruhe Institute of Technology

## Alfred Müller

University of Siegen

Date Written: May 30, 2016

### Abstract

In this paper we introduce the expectile order, defined by X \leq_e Y if e_\alpha(X) \leq e_\alpha(Y) for each \alpha \in (0,1), where e_\alpha denotes the \alpha-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 \leq_{st} and \leq_{cx}, 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 \leq_{st}, \leq_{icx} and \leq_e orders in the family of Lomax distributions.

Keywords: Expectile order, Omega ratio, stop-loss transform, third-order stochastic dominance, skew-normal distribution, Lomax distribution

Suggested Citation

Bellini, Fabio and Klar, Bernhard and Müller, Alfred, Expectiles, Omega Ratios and Stochastic Ordering (May 30, 2016). Available at SSRN: https://ssrn.com/abstract=2788493 or http://dx.doi.org/10.2139/ssrn.2788493