Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 855–873 | Cite as

Expectiles, Omega Ratios and Stochastic Ordering

  • Fabio Bellini
  • Bernhard Klar
  • Alfred Müller


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.


Expectile order Omega ratio Stop-loss transform Third-order stochastic dominance Skew-normal distribution Lomax distribution 

Mathematics Subject Classification (2010)

60E15 60E05 91B82 


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  1. Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228MathSciNetCrossRefzbMATHGoogle Scholar
  2. Azzalini A (1985) A class of distribution which includes the normal ones. Scand J Stat 12:171–178MathSciNetzbMATHGoogle Scholar
  3. Bäuerle N, Müller A (2006) Stochastic Orders and Risk Measures: Consistency and Bounds. Insurance: Mathematics and Economics 38:132–148MathSciNetzbMATHGoogle Scholar
  4. Bellini F (2012) Isotonicity results for generalized quantiles. Statistics and Probability Letters 82:2017–2024MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bellini F, Bignozzi V (2015) On elicitable risk measures. Quant Finan 15:725–733MathSciNetCrossRefGoogle Scholar
  6. Bellini F, Klar B, Müller A, Rosazza Gianin E (2014) Generalized quantiles as risk measures. Insurance: Mathematics and Economics 54:41–48MathSciNetzbMATHGoogle Scholar
  7. Blasi F, Scarlatti S (2012) From Normal vs Skew-Normal Portfolios: FSD and SSD Rules. Journal of Mathematical Finance 2:90–95CrossRefGoogle Scholar
  8. Eilers PHC (2013) Discussion: the beauty of expectiles. Stat Model 13:317–322MathSciNetCrossRefGoogle Scholar
  9. Farooq M, Steinwart I (2015) An SVM-like Approach for Expectile Regression. arXiv:1507.03887
  10. Giles DE, Feng H, Godwin RT (2013) On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution. Communications in Statistics - Theory and Methods 42:1934–1950MathSciNetCrossRefzbMATHGoogle Scholar
  11. Jones MC (1994) Expectiles and M-quantiles are quantiles. Statistics and Probability Letters 20:149–153MathSciNetCrossRefzbMATHGoogle Scholar
  12. Keating C, Shadwick WF (2002) A Universal Performance Measure. The Finance Development Centre, LondonGoogle Scholar
  13. Keating C, Shadwick WF (2002) An Introduction to Omega. The Finance Development Centre, LondonGoogle Scholar
  14. Kneib T (2013) Beyond mean regression. Stat Model 13:275–303MathSciNetCrossRefGoogle Scholar
  15. Koenker R (2013) Discussion: Living beyond our means. Stat Model 13:323–333MathSciNetCrossRefGoogle Scholar
  16. Lopez-Cabrera B, Schulz F (2014) Forecasting generalized quantiles of electricity demand: A functional data approach. SFB 649 Discussion Paper, No. 2014-030. Humboldt University, BerlinGoogle Scholar
  17. Müller A (1996) Ordering of risks: A comparative study via stop-loss transforms. Insurance: Mathematics and Economics 17:215–222MathSciNetzbMATHGoogle Scholar
  18. Müller A (1997) Stochastic Orders generated by Integrals: A Unified Study. Adv Appl Probab 29:414–428MathSciNetCrossRefzbMATHGoogle Scholar
  19. Müller A, Stoyan D (2002) Comparison Methods for Stochastic Models and Risks. John Wiley & Sons Ltd., ChichesterzbMATHGoogle Scholar
  20. Newey K, Powell J (1987) Asymmetric least squares estimation and testing. Econometrica 55:819–847MathSciNetCrossRefzbMATHGoogle Scholar
  21. NOAA (2016) National Centers for Environmental Information (NCEI) U.S. Billion-Dollar Weather and Climate Disasters.
  22. Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131MathSciNetCrossRefzbMATHGoogle Scholar
  23. Remillard B (2013) Statistical Methods for Financial Engineering, Chapman and Hall/CRCGoogle Scholar
  24. Schnabel SK, Eilers PHC (2009) An analysis of life expectancy and economic production using expectile frontier zones. Demogr Res 21:109–134CrossRefGoogle Scholar
  25. Schulze-Waltrup L, Sobotka F, Kneib T, Kauermann G (2015) Expectile and quantile regression: David and Goliath?. Stat Model 15:433–456MathSciNetCrossRefGoogle Scholar
  26. Shaked M, Shantikumar JG (2007) Stochastic Orders, Springer Series in StatisticsGoogle Scholar
  27. Whitmore GA (1970) Third-Degree Stochastic Dominance. American Economic Review 60:457–459Google Scholar
  28. Ziegel JF (2016) Coherence and elicitability. Math Financ 26:901–918MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Dipartimento di Statistica e Metodi QuantitativiUniversità di Milano BicoccaMilanoItaly
  2. 2.Department of MathematicsKarlsruher Institut für Technologie (KIT)KarlsruheGermany
  3. 3.Department of MathematicsUniversity of SiegenSiegenGermany

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