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Bounds on Spearman’s rho when at least one random variable is discrete

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Abstract

Spearman’s rho is one of the most popular dependence measures used in practice to describe the association between two random variables. However, in case of at least one random variable being discrete, Spearman’s correlations are often bounded and restricted to a sub-interval of \([-1,1]\). Hence, small positive values of Spearman’s rho may actually support a strong positive dependence when getting close to its highest attainable value. Similarly, slight negative values of Spearman’s rho can actually mean a strong negative dependence. In this paper, we derive the best-possible upper and lower bounds for Spearman’s rho when at least one random variable is discrete. We illustrate the obtained lower and upper bounds in some situations of practical relevance.

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References

  1. Breiman L, Friedman J, Olshen R, Stone C (1984) Classification and regression trees. Wadsworth, New York

    MATH  Google Scholar 

  2. Capéraà P, Genest C (1993) Spearman’s \(\rho \) is larger than Kendall’s \(\tau \) for positively dependent random variables. J Nonparametric Stat 2(2):183–194

    Article  MathSciNet  Google Scholar 

  3. Daniels HE (1950) Rank correlation and population models. J R Stat Soc Ser B (Methodol) 12(2):171–181

    MathSciNet  MATH  Google Scholar 

  4. Denuit M, Mesfioui M, Trufin J (2019a) Bounds on concordance-based validation statistics in regression models for binary responses. Methodol Comput Appl Probab 21(2):491–509

    Article  MathSciNet  Google Scholar 

  5. Denuit M, Mesfioui M, Trufin J (2019b) Concordance-based predictive measures in regression models for discrete responses. Scand Actuar J 2019(10):824–836

    Article  MathSciNet  Google Scholar 

  6. Dhaene J, Denuit M, Goovaerts M. J, Kaas R, Vyncke D (2002) The concept of comonotonicity in actuarial science and finance: theory. Insurance: Math Econ 31(1): 3–33

  7. Durbin J, Stuart A (1951) Inversions and rank correlation coefficients. J R Stat Soc Ser B (Methodol) 13(2):303–309

    MathSciNet  MATH  Google Scholar 

  8. Garrido J, Genest C, Schulz J (2016) Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Math Econ 70: 205–215

  9. Gschlossl S, Czado C (2007) Spatial modelling of claim frequency and claim size in non-life insurance. Scand Actuar J 2007(3):202–225

    Article  MathSciNet  Google Scholar 

  10. Hastie T (2017) gam: Generalized Additive Models. https://CRAN.R-project.org/package=gam. R package version 1.14-4

  11. Mesfioui M, Tajar A (2005) On the properties of some nonparametric concordance measures in the discrete case. Nonparametric Stat 17:541–554

    Article  MathSciNet  Google Scholar 

  12. Nešlehová J (2007) On rank correlation measures for non-continuous random variables. J Multivar Anal 98:544–567

    Article  MathSciNet  Google Scholar 

  13. Pechon F (2019) rfCountData. https://github.com/fpechon/rfCountData

  14. Schreyer M, Paulin R, Trutschnig W (2017) On the exact region determined by Kendall’s \(\tau \) and Spearman’s \(\rho \). J R Stat Soc Ser B (Stat Methodol) 79(2):613–633

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Mhamed Mesfioui acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada No 06536-2018.

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Authors and Affiliations

  1. Département de mathématiques et d’informatique, Université du Québec à Trois-Rivières, Trois-Rivières, QC, G9A 5H7, Canada

    Mhamed Mesfioui

  2. Department of Mathematics, Université Libre de Bruxelles (ULB), Brussels, Belgium

    Julien Trufin & Pierre Zuyderhoff

Authors
  1. Mhamed Mesfioui
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  2. Julien Trufin
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  3. Pierre Zuyderhoff
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Correspondence to Julien Trufin.

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Mesfioui, M., Trufin, J. & Zuyderhoff, P. Bounds on Spearman’s rho when at least one random variable is discrete. Eur. Actuar. J. 12, 321–348 (2022). https://doi.org/10.1007/s13385-021-00289-8

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  • DOI: https://doi.org/10.1007/s13385-021-00289-8

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