Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions

  • Antonio Di CrescenzoEmail author
  • Barbara Martinucci
  • Julio Mulero
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10672)


Distorted distributions were introduced in the context of actuarial science for several variety of insurance problems. In this paper we consider the quantile-based probabilistic mean value theorem given in Di Crescenzo et al. [4] and provide some applications based on distorted random variables. Specifically, we consider the cases when the underlying random variables satisfy the proportional hazard rate model and the proportional reversed hazard rate model. A setting based on random variables having the ‘new better than used’ property is also analyzed.


Quantile function Distorted distribution Mean value theorem 



The research of A. Di Crescenzo and B. Martinucci has been performed under partial support by the Group GNCS of INdAM. J. Mulero acknowledges support received from the Ministerio de Economía, Industria y Competitividad under grant MTM2016-79943-P (AEI/FEDER, UE).


  1. 1.
    Balbás, A., Garrido, J., Mayoral, S.: Properties of distortion risk measures. Methodol. Comput. Appl. Probab. 11, 385–399 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Denneberg, D.: Premium calculation: why standard deviation should be replaced by absolute deviation. ASTIN Bull. 20, 181–190 (1990)CrossRefGoogle Scholar
  3. 3.
    Di Crescenzo, A.: A probabilistic analogue of the mean value theorem and its applications to reliability theory. J. Appl. Probab. 36, 706–719 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Di Crescenzo, A., Martinucci, B., Mulero, J.: A quantile-based probabilistic mean value theorem. Probab. Eng. Inf. Sci. 30, 261–280 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Crescenzo, A., Meoli, A.: On the fractional probabilistic Taylor’s and mean value theorems. Fract. Calc. Appl. Anal. 19, 921–939 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gupta, N., Misra, N., Kumar, S.: Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components. Eur. J. Oper. Res. 240, 425–430 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lin, G.D.: On a probabilistic generalization of Taylor’s theorem. Stat. Prob. Lett. 19, 239–243 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Massey, W.A., Whitt, W.: A probabilistic generalization of Taylor’s theorem. Stat. Prob. Lett. 16, 51–54 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Navarro, J., del Águila, Y., Sordo, M.A., Suárez-Llorens, A.: Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodol. Comput. Appl. Prob. 18, 529–545 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Quiggin, J.: A theory of anticipated utility. J. Econ. Behav. Organ. 3, 323–343 (1982)CrossRefGoogle Scholar
  11. 11.
    Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sordo, M.A., Navarro, J., Sarabia, J.M.: Distorted Lorenz curves: models and comparisons. Soc. Choice Welf. 42, 761–780 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sordo, M.A., Suárez-Llorens, A.: Stochastic comparisons of distorted variability measures. Insur. Math. Econ. 49, 11–17 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sordo, M.A., Suárez-Llorens, A., Bello, A.J.: Comparison of conditional distributions in portfolios of dependent risks. Insur. Math. Econ. 61, 62–69 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, S.: Insurance pricing and increased limits ratemaking by proportional Hazards transforms. Insur. Math. Econ. 17, 43–54 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, S.: Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 71–92 (1996)CrossRefGoogle Scholar
  17. 17.
    Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Antonio Di Crescenzo
    • 1
    Email author
  • Barbara Martinucci
    • 1
  • Julio Mulero
    • 2
  1. 1.Dipartimento di MatematicaUniversità di SalernoFiscianoItaly
  2. 2.Departamento de MatemáticasUniversidad de AlicanteAlicanteSpain

Personalised recommendations