Skip to main content

On the Skewness Order of van Zwet and Oja


Van Zwet (1964) [16] introduced the convex transformation order between two distribution functions F and G, defined by FcG if G−1F is convex. A distribution which precedes G in this order should be seen as less right-skewed than G. Consequently, if FcG, any reasonable measure of skewness should be smaller for F than for G. This property is the key property when defining any skewness measure.

In the existing literature, the treatment of the convex transformation order is restricted to the class of differentiable distribution functions with positive density on the support of F. It is the aim of this work to analyze this order in more detail. We show that several of the most well known skewness measures satisfy the key property mentioned above with very weak or no assumptions on the underlying distributions. In doing so, we conversely explore what restrictions are imposed on the underlying distributions by the requirement that F precedes G in convex transformation order.

This is a preview of subscription content, access via your institution.


  1. B. C. Arnold and R. A. Groeneveld, “Skewness and Kurtosis Orderings: an Introduction”, in IMS Lecture Notes — Monograph Series, Vol. 22: Stochastic Inequalities (IMS, Hayward, CA, 1993), pp. 17–24.

    Google Scholar 

  2. B. C. Arnold and R. A. Groeneveld, “Measuring Skewness with Respect to the Mode”, Amer. Statist. 49, 34–38 (1995).

    MathSciNet  Google Scholar 

  3. E. Artin, The Gamma Function (New York—Chicago—San Francisco—Toronto—London, Holt, Rinehart and Winston, 1964).

    MATH  Google Scholar 

  4. A, L. Bowley, Elements of Statistics (London, P. S. King & Son, 1901), 1st ed.

    MATH  Google Scholar 

  5. R. A. Groeneveld and G. Meeden, “Measuring Skewness and Kurtosis”, J. Roy.Statist. Soc. Ser. D (The Statistician) 33, 319–399 (1984).

    Google Scholar 

  6. D. V. Hinkley, “On Power Transformations to Symmetry”, Biometrika 62, 101–111 (1975).

    MathSciNet  Article  Google Scholar 

  7. H. Hotelling and L. M. Solomons, “The Limits of a Measure of Skewness”, Ann. Math. Statist. 3, 141–142 (1932).

    Article  Google Scholar 

  8. H. L. MacGillivray, “Skewness and Asymmetry: Measures and Orderings”, Ann. Statist. 14, 994–1011 (1986).

    MathSciNet  Article  Google Scholar 

  9. K. N. Majindar, “Improved Bounds on a Measure of Skewness”, Ann. Math. Statist. 33, 1192–1194 (1962).

    MathSciNet  Article  Google Scholar 

  10. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications. A Contemporary Approach, in CMS Books in Mathematics (Springer, 2018), 2nd ed.

  11. H. Oja, “On Location, Scale, Skewness and Kurtosis of Univariate Distributions”, Scand. J. Statist. 8, 154–168 (1981).

    MathSciNet  MATH  Google Scholar 

  12. K. Pearson, “Contributions to the Mathematical Theory of Evolution, II. Skew Variation in Homogeneous Material”, Philos. Trans. Roy. Soc., London. A 186, 343–414 (1895).

    Article  Google Scholar 

  13. M. Shaked and G. J. Shanthikumar, Stochastic Orders (Springer, 2006).

  14. G. R. Shorak and J. A. Wellner, Empirical Processes with Applications to Statistics (Wiley, 1986).

  15. G. U. Yule, An Introduction to the Theory of Statistics (Charles Griffin & Co., Ltd., 1922), 6th ed.

  16. W. R. van Zwet, Convex Transformations of Random Variables in Mathematical Centre Tracts (Amsterdam, 1964).

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. Eberl.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Eberl, A., Klar, B. On the Skewness Order of van Zwet and Oja. Math. Meth. Stat. 28, 262–278 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • convex order
  • convex transformation order
  • measure of skewness
  • quantile skewness

AMS 2010 Subject Classification

  • 60E05
  • 60E15
  • 62E10