On the bivariate Sarmanov distribution and copula. An application on insurance data using truncated marginal distributions

• Zuhair Bahraoui ^[1] ; Catalina Bolancé ^[1] ; Elena Pelican ^[2] ; Raluca Vernic ^[2]
  1. [1] Universitat de Barcelona

     Universitat de Barcelona

     Barcelona, España

     
  2. [2] Universitatea Ovidius din Constanta
• Localización: Sort: Statistics and Operations Research Transactions, ISSN 1696-2281, Vol. 39, Nº. 2, 2015, págs. 209-230
• Idioma: inglés
• Enlaces
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• Resumen

  • The Sarmanov family of distributions can provide a good model for bivariate random variables and it is used to model dependency in a multivariate setting with given marginals. In this paper, we focus our attention on the bivariate Sarmanov distribution and copula with different truncated extreme value marginal distributions. We compare a global estimation method based on maximizing the full log-likelihood function with the estimation based on maximizing the pseudo-log-likelihood function for copula (or partial estimation). Our aim is to estimate two statistics that can be used to evaluate the risk of the sum exceeding a given value. Numerical results using a real data set from the motor insurance sector are presented.

• Referencias bibliográficas
  • Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.
  • Bahraoui, Z., Bolancé, C. and Pérez-Marı́n, M. (2014). Testing extreme value copulas to estimate the quantile. SORT-Statistics and Operations...
  • Bairamov, I., Altinsoy, B. and Kerns, G. (2011). On generalized Sarmanov bivariate distributions. TWMS Journal of Applied and Engineering...
  • Bairamov, I., Kotz, S. and Gebizlioglu, O. L. (2001). The Sarmanov family and its generalization: theory and methods. South African Statistical...
  • Bolancé, C., Bahraoui, Z. and Artı́s, M. (2014). Quantifying the risk using copulae with nonparametric marginals. Insurance: Mathematics...
  • Bolancé, C., Guillén, M., Pelican, E. and Vernic, R. (2008). Skewed bivariate models and non-parametric estimation for CTE risk measure....
  • Bourden, R. and Faires, J. (2001). Numerical Analysis. 7th Edition. Brooks/Cole, Pacific Grove, CA.
  • Buch-Larsen, T., Guillén, M., Nielsen, J. and Bolancé, C. (2005). Kernel density estimation for heavy-tailed distributions using the Champernowne...
  • Champernowne, D. (1952). The graduation of income distribution. Econometrica, 20, 591–615.
  • Cohen, L. (1984). Probability distributions with given multivariate marginals. Journal of Mathematical Physics, 25, 2402–2403.
  • Danaher, P. (2007). Modeling page views across multiple websites with an application to internet reach and frequency prediction. Marketing...
  • Farlie, D. (1960). The performance of some correlation coeficients for a general bivariate distribution. Biometrika, 47, 307–323.
  • Fisk, P. (1961). The graduation of income distribution. Econometrica, 29, 171–185.
  • Gómez-Déniz, E. and Pérez-Rodrı́guez, J. (2015). Closed form solution for a bivariate distribution in stochastic frontier models with...
  • Hashorva, E. and Ratovomirija, G. (2015). On Sarmanov mixed Erlang risks in insurance applications. ASTIN Bulletin, 45, 175–205.
  • Hernández-Bastida, A. and Fernández-Sánchez, M. (2013). A Sarmanov family with beta and gamma marginal distributions: an application...
  • Hernández-Bastida, A., Fernández-Sánchez, M. and Gómez-Déniz, E. (2009). The net Bayes premium with dependence between the risk...
  • Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, New York.
  • Kotz, S., Balakrishnan, N. and Johnson, N. (2000). Continuous Multivariate Distributions. Vol.1: Models and Applications. John Wiley &...
  • Lee, M. (1996). Properties and applications of the Sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods,...
  • Mladenovic, N. and Hansen, P. (1997). Variable neighborhood search. Computers and Operations Research, 24, 1097–1100.
  • Nelsen, R. (2006). An Introduction to Copulas. 2nd ed. Springer, New York.
  • Pelican, E. and Vernic, R. (2013). Maximum-likelihood estimation for the multivariate Sarmanov distribution: simulation study. International...
  • Sarmanov, O. (1966). Generalized normal correlation and two-dimensional Frechet classes.Doclady (Soviet Mathematics), 168, 596–599.
  • Shubina, M. and Lee, T. (2004). On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions....
  • Vernic, R. (2014). On the distribution of a sum of Sarmanov distributed random variables. Journal of Theoretical Probability, Forthcoming.
  • Yang, Y. and Hashorva, E. (2013). Extremes and products of multivariate AC-product risks. Insurance: Mathematics and Economics, 52, 312–319.