Stochastic ordering of variability measure estimators

Abstract

Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples.

Keywords:

• Stochastic orders
• variability measures
• sample variance
• Gini mean difference

AMS 2020 Subject Classifications:

• Primary 60E15
• 62D99 Secondary 94A20

Acknowledgments

We sincerely thanks the two reviewers and the Associate Editor who made comments and suggestions that significantly contributed to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

J. Baz is partially supported by Programa Severo Ochoa of Principality of Asturias (BP21042). F. Pellerey carried out this study within the FAIR - Future Artificial Intelligence Research and received funding from the European Union Next-GenerationEU (Piano Nazionale di Ripresa e Resilienza (PNRR) Missione 4 Componente 2, Investimento 1.3 D.D. 1555 11/10/2022, PE00000013). J. Baz, S. Montes and I. Díaz are been supported by the Ministry of Science and Innovation (PDI2022-139886NB-l00).