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A Sarmanov family with beta and gamma marginal distributions: an application to the Bayes premium in a collective risk model

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  • Agustín Hernández-Bastida

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  • M. Fernández-Sánchez

Abstract

In this paper we firstly develop a Sarmanov–Lee bivariate family of distributions with the beta and gamma as marginal distributions. We obtain the linear correlation coefficient showing that, although it is not a strong family of correlation, it can be greater than the value of this coefficient in the Farlie–Gumbel–Morgenstern family. We also determine other measures for this family: the coefficient of median concordance and the relative entropy, which are analyzed by comparison with the case of independence. Secondly, we consider the problem of premium calculation in a Poisson–Lindley and exponential collective risk model, where the Sarmanov–Lee family is used as a structure function. We determine the collective and Bayes premiums whose values are analyzed when independence and dependence between the risk profiles are considered, obtaining that notable variations in premiums values are obtained even when low levels of correlation are considered. Copyright Springer-Verlag 2012

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Bibliographic Info

Article provided by Springer in its journal Statistical Methods & Applications.

Volume (Year): 21 (2012)
Issue (Month): 4 (November)
Pages: 391-409

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Handle: RePEc:spr:stmapp:v:21:y:2012:i:4:p:391-409

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Related research

Keywords: Aggregate loss distribution; Bayesian analysis; Structure function; Poisson–Lindley distribution; Divergence;

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  1. David A. Schweidel & Peter S. Fader & Eric T. Bradlow, 2008. "A Bivariate Timing Model of Customer Acquisition and Retention," Marketing Science, INFORMS, vol. 27(5), pages 829-843, 09-10.
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  3. Martel-Escobar, M. & Hernández-Bastida, A. & Vázquez-Polo, F.J., 2012. "On the independence between risk profiles in the compound collective risk actuarial model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1419-1431.
  4. Samuel Kotz & J. Renevan Dorp, 2002. "A versatile bivariate distribution on a bounded domain: Another look at the product moment correlation," Journal of Applied Statistics, Taylor & Francis Journals, vol. 29(8), pages 1165-1179.
  5. Heilmann, Wolf-Rudiger, 1989. "Decision theoretic foundations of credibility theory," Insurance: Mathematics and Economics, Elsevier, vol. 8(1), pages 77-95, March.
  6. Woojune Yi & Vicki M. Bier, 1998. "An Application of Copulas to Accident Precursor Analysis," Management Science, INFORMS, vol. 44(12-Part-2), pages S257-S270, December.
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