Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-04T00:29:42.025Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SOME PROPERTIES OF A CLASS OF MULTIVARIATE ERLANG MIXTURES WITH INSURANCE APPLICATIONS

Published online by Cambridge University Press:  23 September 2014

Gordon E. Willmot  and
Jae-Kyung Woo
Gordon E. Willmot
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Canada E-mail: gewillmo@math.uwaterloo.ca
Jae-Kyung Woo*
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
*
E-mail: jkwoo@hku.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

Keywords

Multivariate mixed Erlangscale mixturesgeneralized Esscher transformationstop-loss momentsresidual lifetime distributionrisk measurescapital allocationjoint and last survivor
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 1 , January 2015 , pp. 151 - 173
Copyright
Copyright © ASTIN Bulletin 2014 

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

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of Risk. Mathematical Finance, 9, 203–228.Google Scholar
Asimit, A.V., Furman, E. and Vernic, R. (2010) On a multivariate Pareto distribution. Insurance: Mathematics and Economics, 46, 308–316.Google Scholar
Assaf, D., Landberg, N., Savits, T. and Shaked, M. (1984) Multivariate phase-type distributions. Operations Research, 32 (3), 688–702.Google Scholar
Bargès, M., Cossette, H. and Marceau, É. (2009) TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics, 45, 348–361.Google Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11, 52–60.Google Scholar
Cai, J. and Li, H. (2005a) Multivariate risk model of phase type. Insurance: Mathematics and Economics, 36, 137–152.Google Scholar
Cai, J. and Li, H. (2005b) Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability, 42 (3), 810–825.Google Scholar
Chiragiev, A. and Landsman, Z. (2007) Multivariate Pareto portfolios: TCE-based capital allocation and divided differences. Scandinavian Actuarial Journal, 4, 261–280.Google Scholar
Cossette, H., Côté, M.-P., Marceau, É. and Moutanabbir, K. (2013) Multivariate distribution defined with Farlie-Gumble-Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation. Insurance: Mathematics and Economics, 52, 560–572.Google Scholar
Cossette, H., Mailhot, M. and Marceau, É. (2012) TVaR-based capital allocation for multivariate compound distribution with positive continuous claim amounts. Insurance: Mathematics and Economics, 50, 247–256.Google Scholar
Denault, M. (2001) Coherent allocation of risk capital. Journal of Risk, 4 (1), 7–21.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. John Wiley & Sons.Google Scholar
Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A. and Vanduffel, S. (2008) Some results on the CTE-based capital allocation rule. Insurance: Mathematics and Economics, 42 (2), 855–863.Google Scholar
Embrechts, P., Maejima, M. and Teugels, J. (1985) Asymptotic behaviour of compound distributions. ASTIN Bulletin, 15, 45–48.Google Scholar
Franco, M. and Vivo, J.-M. (2009) A Multivariate lifetime model based on generalized exponential distributions. Applied Stochastic Models and Data Analysis. The XIII International Conference, Lithuania 378–381.Google Scholar
Furman, E. and Landsman, Z. (2005) Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics, 37 (3), 635–649.Google Scholar
Furman, E. and Landsman, Z. (2006) Tail variance premium with applications for elliptical portfolio of risks. Astin Bulletin, 36 (2), 433–462.Google Scholar
Furman, E. and Landsman, Z. (2008) Economic capital allocations for non-negative portfolios of dependent risks. ASTIN Bulletin, 38 (2), 601–619.Google Scholar
Furman, E. and Landsman, Z. (2010) Multivariate Tweedie distributions and some related capital-at-risk analyses. Insurance: Mathematics and Economics, 46, 351–361.Google Scholar
Furman, E. and Zitikis, R. (2008) Weighted premium calculation principles. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
Gerber, H. (1980) Credibility for Esscher premiums. Mitteilungen der Vereinigung schweiz. Versicherungsmathematiker, 307–312.Google Scholar
Goovaerts, M.J., Kaas, R. and Laeven, R.J.A. (2010) Decision principles derived from risk measures. Insurance: Mathematics and Economics, 47:294–302.Google Scholar
Grandell, J. (1997) Mixed Poisson Processes. London: Chapman & Hall.Google Scholar
Gupta, P.L. and Gupta, R.C. (2001) Failure rate of the minimum and maximum of a multivariate normal distribution. Metrika, 53, 39–49.CrossRefGoogle Scholar
Heilmann, W.R. (1989) Decision theoretic foundations of credibility theory. Insurance: Mathematics and Economics, 8, 77–95.Google Scholar
Hesselager, O. and Andersson, U. (2002) Risk Sharing and Capital Allocation. Tryg Insurance.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. London: Chapman & Hall.Google Scholar
Kijima, M. (2006) A multivariate extension of equilibrum pricing transforms. The multivariate Esscher and Wang transforms for pricing financial and insurance risks. ASTIN Bulletin, 36 (1), 269–283.Google Scholar
Kotz, S., Balakrishman, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions. New York: John Wiley & Sons, Inc.Google Scholar
Kulkarni, V.G. (1989) A new class of multivariate phase type distributions. Operations Research, 37 (1), 151–158.Google Scholar
Lee, S.C.K. and Lin, X.S. (2012) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin, 42 (1), 153–180.Google Scholar
Mathai, A.M. and Moschopoulos, P.G. (1991) On a multivariate gamma”, Journal of Multivariate Analysis, 39, 135–153.Google Scholar
Panjer, H.H. (2002) Measurement of risk, solvency requirements, and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo Research Report, 1–15.Google Scholar
Patil, G.P. and Ord, J.K. (1976) On size-biased sampling and related form-invariant weighted distributions. Sankhyā, Series B, 38, 48–61.Google Scholar
Willmot, G.E. (1989) Limiting tail behaviour of some discrete compound distributions. Insurance: Mathematics and Economics, 8 (3), 175–185.Google Scholar
Willmot, G.E. (2007) On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics, 41, 17–31.Google Scholar
Willmot, G.E., Drekic, S. and Cai, J. (2005) Equilibrium compound distributions and stop-loss moments. Scandinavian Actuarial Journal, 1, 6–24.Google Scholar
Willmot, G.E. and Lin, X.S. (2011) Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27, 2–16.Google Scholar
Willmot, G.E. and Woo, J.-K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–118.Google Scholar