, Volume 82, Issue 1, pp 99–124 | Cite as

Shape measures based on the convex transform order

  • A. ArriazaEmail author
  • A. Di Crescenzo
  • M. A. Sordo
  • A. Suárez-Llorens


Three functional measures of the shape of univariate distributions are proposed which are consistent with respect to the convex transform order. The first two are weighted tail indices that characterize location-scale families of distributions, whilst the third is a skewness measure. Properties of the new measures are established for various classes of symmetric and asymmetric distributions, and the generalized Pareto distribution characterized in terms of them. Kernel density based estimation of the measures is also considered, and the use of the estimated functionals is illustrated in the analysis of two real data sets.


Convex transform order Kurtosis Shape Skewness Stochastic order Tail weight Generalized Pareto distribution 



The authors would like to thank two anonymous referees and the Editor of the journal for their extensive and helpful comments which have led to significant improvements in both the content and presentation of the paper. The authors also acknowledge support received from the Ministerio de Economía y Competitividad (Spain) under Grants MTM2014-57559-P and MTM2017-89577-P, and from the group GNCS of INdAM.

Compliance with ethical standards

Conflict of interest

On behalf of all the authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dpto. Estadística e Investigación Operativa, Facultad de CienciasUniversidad de CádizPuerto RealSpain
  2. 2.Dipartimento di MatematicaUniversità di SalernoFiscianoItaly

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