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On the distribution of the (un)bounded sum of random variables

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  • Cherubini, Umberto
  • Mulinacci, Sabrina
  • Romagnoli, Silvia

Abstract

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.

Suggested Citation

  • Cherubini, Umberto & Mulinacci, Sabrina & Romagnoli, Silvia, 2011. "On the distribution of the (un)bounded sum of random variables," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 56-63, January.
    • Handle: RePEc:eee:insuma:v:48:y:2011:i:1:p:56-63
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    References listed on IDEAS

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    1. Denuit, M. & Genest, C. & Marceau, E., 1999. "Stochastic bounds on sums of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 85-104, September.
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    Cited by:

    1. Jorge Navarro & Franco Pellerey & Julio Mulero, 2022. "On sums of dependent random lifetimes under the time-transformed exponential model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 879-900, December.
    2. Gijbels, Irène & Herrmann, Klaus, 2014. "On the distribution of sums of random variables with copula-induced dependence," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 27-44.
    3. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.

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