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

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Computer Aided Systems Theory – EUROCAST 2017 (EUROCAST 2017)

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Abstract

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.

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References

  1. Balbás, A., Garrido, J., Mayoral, S.: Properties of distortion risk measures. Methodol. Comput. Appl. Probab. 11, 385–399 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Denneberg, D.: Premium calculation: why standard deviation should be replaced by absolute deviation. ASTIN Bull. 20, 181–190 (1990)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Di Crescenzo, A., Martinucci, B., Mulero, J.: A quantile-based probabilistic mean value theorem. Probab. Eng. Inf. Sci. 30, 261–280 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Crescenzo, A., Meoli, A.: On the fractional probabilistic Taylor’s and mean value theorems. Fract. Calc. Appl. Anal. 19, 921–939 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin, G.D.: On a probabilistic generalization of Taylor’s theorem. Stat. Prob. Lett. 19, 239–243 (1994)

    Article  MathSciNet  Google Scholar 

  8. Massey, W.A., Whitt, W.: A probabilistic generalization of Taylor’s theorem. Stat. Prob. Lett. 16, 51–54 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Quiggin, J.: A theory of anticipated utility. J. Econ. Behav. Organ. 3, 323–343 (1982)

    Article  Google Scholar 

  11. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sordo, M.A., Navarro, J., Sarabia, J.M.: Distorted Lorenz curves: models and comparisons. Soc. Choice Welf. 42, 761–780 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sordo, M.A., Suárez-Llorens, A.: Stochastic comparisons of distorted variability measures. Insur. Math. Econ. 49, 11–17 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, S.: Insurance pricing and increased limits ratemaking by proportional Hazards transforms. Insur. Math. Econ. 17, 43–54 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, S.: Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 71–92 (1996)

    Article  Google Scholar 

  17. Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Salerno, Fisciano, Italy

    Antonio Di Crescenzo & Barbara Martinucci

  2. Departamento de Matemáticas, Universidad de Alicante, Alicante, Spain

    Julio Mulero

Authors
  1. Antonio Di Crescenzo
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  2. Barbara Martinucci
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  3. Julio Mulero
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Corresponding author

Correspondence to Antonio Di Crescenzo .

Editor information

Editors and Affiliations

  1. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. Johannes Kepler University Linz, Linz, Austria

    Franz Pichler

  3. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Alexis Quesada-Arencibia

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Di Crescenzo, A., Martinucci, B., Mulero, J. (2018). Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10672. Springer, Cham. https://doi.org/10.1007/978-3-319-74727-9_10

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  • DOI: https://doi.org/10.1007/978-3-319-74727-9_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74726-2

  • Online ISBN: 978-3-319-74727-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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