Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-03T06:03:32.126Z Has data issue: false hasContentIssue false
  • English
  • Français

AGGREGATION OF DEPENDENT RISKS IN MIXTURES OF EXPONENTIAL DISTRIBUTIONS AND EXTENSIONS

Published online by Cambridge University Press:  25 April 2018

José María Sarabia ,
Emilio Gómez-Déniz ,
Faustino Prieto  and
Vanesa Jordá
José María Sarabia*
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain
Emilio Gómez-Déniz
Affiliation:
Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, 35017 Las Palmas de G.C., Spain E-Mail: emilio.gomez-deniz@ulpgc.es
Faustino Prieto
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain E-Mail: faustino.prieto@unican.es
Vanesa Jordá
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain E-Mail: vanesa.jorda@unican.es
*
E-Mail: sarabiaj@unican.es
Get access Rights & Permissions [Opens in a new window]

Abstract

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modelled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al., 2016), gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

Keywords

Aggregationdependent random variablesLaplace transformcollective risk modelpartial Bell polynomials
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 1079 - 1107
Copyright
Copyright © Astin Bulletin 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I.A. (1970) Handbook of Mathematical Functions. New York: Dover Publications. Inc.Google Scholar
Albrecher, H., Constantinescu, C. and Loisel, S. (2011) Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics, 48, 265–270.Google Scholar
Albrecher, H. and Kortschak, D. (2009) On ruin probability and aggregate claim representations for Pareto claim size distributions. Insurance: Mathematics and Economics, 45, 362–373.Google Scholar
Arbenz, P., Hummel, C. and Mainik, G. (2012) Copula based hierarchical risk aggregation through sample reordering. Insurance: Mathematics and Economics, 51, 122–133.Google Scholar
Arnold, B.C. (1983) Pareto Distributions. Fairland, MD: International Cooperative Publishing House.Google Scholar
Arnold, B.C. (2015) Pareto Distributions, 2nd ed. Boca Ratón, FL: Chapman & Hall/CRC Monographs on Statistics & Applied Probability.Google Scholar
Asgharzadeh, A., Nadarajah, S. and Sharafi, F. (2017) Generalized inverse Lindley distribution with application to Danish fire insurance data. Communications in Statistics-Theory and Methods, 46 (10), 5001–5021.Google Scholar
Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities, 2nd ed. New Jersey: World Scientific.Google Scholar
Barlow, R.E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston.Google Scholar
Bølviken, E. and Guillen, M. (2017) Risk aggregation in Solvency II through recursive log-normals. Insurance: Mathematics and Economics, 73, 20–26.Google Scholar
Coqueret, G. (2014) Second order risk aggregation with the Bernstein copula. Insurance: Mathematics and Economics, 58, 150–158.Google Scholar
Cossette, H., Côté, M.P., Marceau, E. and Moutanabbir, K. (2013) Multivariate distribution defined with Farlie–Gumbel–Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation. Insurance: Mathematics and Economics, 52, 560–572.Google Scholar
Côté, M.P. and Genest, C. (2015) A copula-based risk aggregation model. The Canadian Journal of Statistics, 43, 60–81.Google Scholar
Dacarogna, M., Elbahtouri, L. and Kratz, M. (2015) Explicit diversification benefit for dependent risks. Working paper, ESSEC Business School.Google Scholar
Esary, J.D., Proschan, F. and Walkup, D.W. (1967) Association of random variables, with applications. Annals of Mathematical Statistics, 38, 1466–1474.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 1, 2nd ed. New York: John Wiley.Google Scholar
Genest, C. and MacKay, J. (1986) The joy of copulas: bivariate distributions with uniform marginals. The American Statistician, 40, 280–283.Google Scholar
Gijbels, I. and Herrmann, K. (2014) On the distribution of sums of random variables with copula-induced dependence. Insurance: Mathematics and Economics, 59, 27–44.Google Scholar
Gleser, L.J. (1989) The gamma distribution as a mixture of exponential distributions. The American Statistician, 43, 115–117.Google Scholar
Gómez–Déniz, E., Sarabia, J.M. and Balakrishnan, N. (2012) A multivariate discrete Poisson-Lindley distribution: Extensions and actuarial applications. ASTIN Bulletin, 42 (2), 655–678.Google Scholar
Good, I.J. (1953) The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237–260.Google Scholar
Goovaerts, M., Kaas, R., Laeven, R., Tang, Q. and Vernic, R. (2005) The tail probability of discounted sums of Pareto-like losses in insurance. Scandinavian Actuarial Journal, 6, 446–461.Google Scholar
Gradshteyn, L.S. and Ryzhik, I. M. (1980) Table of Integrals, Series and Products. New York: Academic Press.Google Scholar
Guillén, M., Sarabia, J.M. and Prieto, F. (2013) Simple risk measure calculations for sums of positive random variables. Insurance: Mathematics and Economics, 53, 273–280.Google Scholar
Jewell, N.P. (1982) Mixtures of exponential distributions. The Annals of Statistics, 10, 479–484.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. London: Chapman & Hall.Google Scholar
Hashorva, E. and Ratovomirija, G. (2015) On Sarmanov mixed Erlang risks in insurance applications. ASTIN Bulletin, 45, 175–205.Google Scholar
Hougaard, P. (1986) Survival models for heterogeneous populations derived from stable distributions. Biometrika, 73, 387–396.Google Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 1, 2nd ed., New York: John Wiley.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss Models. From Data to Decisions, 3rd ed. New York: John Wiley.Google Scholar
Krantz, S.G. and Parks, H.R. (2002) A Primer of Real Analytic Functions, Birkhäuser Advanced Texts - Basler Lehrbücher, 2nd ed. Boston: Birkhäuser Verlag.Google Scholar
Lee, M.-L.T. and Gross, A.J. (1989) Properties of conditionally independent generalized gamma distributions. Probability in the Engineering and Informational Sciences, 3, 289–297.Google Scholar
Lindley, D.V. (1958) Fiducial distributions and Bayes's theorem. Journal of the Royal Statistical Society: Series B, 20, 102–107.Google Scholar
Lindley, D.V. and Singpurwalla, N.D. (1986) Multivariate distributions for the life lengths of components of a system sharing a common environment. Journal of Applied Probability, 23, 418–431.Google Scholar
McDonald, J.B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.Google Scholar
Nayak, T.K. (1987) Multivariate Lomax distribution: Properties and usefulness in reliability theory. Journal of Applied Probability, 24, 170–177.Google Scholar
Nelsen, R. (1999) An Introduction to Copulas. New York: Springer.Google Scholar
Oakes, D. (1989) Bivariate survival models induced by frailties. Journal of the American Statistical Association, 84, 487–493.Google Scholar
Roehner, B. and Winniwarter, P. (1985) Aggregation of independent Paretian random variables. Advances in Applied Probability, 17, 465–469.Google Scholar
Roy, D. and Mukherjee, S.P. (1988) Generalized mixtures of exponential distributions. Journal of Applied Probability, 25, 510–518.Google Scholar
Sarabia, J.M., Gómez-Déniz, E., Prieto, F. and Jordá, V. (2016) Risk aggregation in multivariate dependent Pareto distributions. Insurance: Mathematics and Economics, 71, 154–163.Google Scholar
Sibuya, M. (1979) Generalized Hypergeometric, Digamma and Trigamma Distributions. Annals of the Institute of Statistical Mathematics, 31, 373–390.Google Scholar
Vernic, R. (2016) On the distribution of a sum of Sarmanov distributed random variables. Journal of Theoretical Probability, 29, 118–142.Google Scholar
Watson, G.N. (1995) A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, UK: Cambridge University Press.Google Scholar
Whitmore, G.A. (1988) Inverse Gaussian mixtures of exponential distributions. McGill University, Faculty of Management, unpublished paper.Google Scholar
Whitmore, G.A. and Lee, M.-L.T. (1991) A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials. Technometrics, 33, 39–50.Google Scholar