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Discrete-Time Risk Models Based on Time Series for Count Random Variables

Published online by Cambridge University Press:  09 August 2013

Hélène Cossette
Affiliation:
École d'Actuariat, Université Laval, Québec, Canada
Etienne Marceau
Affiliation:
École d'Actuariat, Université Laval, Québec, Canada
Véronique Maume-Deschamps
Affiliation:
Université de Lyon, Université Lyon 1, ISFA, Laboratoire SAF

Abstract

In this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

Type
Research Article
Copyright
Copyright © International Actuarial Association 2010

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References

Al-Osh, M.A. and Alzaid, A.A. (1987) First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8, 261275.CrossRefGoogle Scholar
Al-Osh, M.A. and Alzaid, A.A. (1988) Integer-valued moving average (INMA) process. Statistical Papers 29, 281300.CrossRefGoogle Scholar
Arvidsson, H. and Francke, S. (2007) Dependence in non life insurance. U.U.D.M Project Report 2007:23. Department of Mathematics, Uppsala University (http://www.math.uu.se/research/pub/Arvidsson/Francke1.pdf).Google Scholar
Brännäs, K., Hellström, J. and Nordström;, J. (2002) A new approach to modelling and forecasting monthly guest nights in hotels. International Journal of Forecasting 18(1), 1930.CrossRefGoogle Scholar
Brannas, K. and Quoreshi, S. (2004) Integer-valued moving average modelling of the number of transactions in stocks. Umea Economic Studies 637.Google Scholar
Bühlmann, H. (1970). Mathematical methods in risk theory. Springer-Verlag, New York.Google Scholar
Cameron, A.C. and Trivedi, P.K. (1998) Regression Analysis of Count Data. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Christ, R. and Steinebach, J. (1995) Estimating the adjustment coefficient in an ARMA(p, q) risk model. Insurance: Mathematics and Economics 17, 149161.Google Scholar
Cossette, H., Landriault, D. and Marceau, E. (2003) Ruin probabilities in the compound Markov binomial model. Scandinavian Actuarial Journal, 301323.CrossRefGoogle Scholar
Cossette, H., Landriault, D. and Marceau, E. (2004a) Exact expressions and upper bound for ruin probabilities in the compound Markov binomial model. Insurance: Mathematics and Economics 34, 449466.Google Scholar
Cossette, H., Landriault, D. and Marceau, E. (2004b) The compound binomial model defined in a Markovian environment. Insurance: Mathematics and Economics 35, 425443.Google Scholar
Cossette, H., Landriault, D. and Marceau, E. (2004c) Ruin measures related to the surplus process in the compound Markov binomial model. Bulletin of the Swiss Association of Actuaries, 77114.Google Scholar
Cossette, H., Marceau, E. and Maume-Deschamps, V. (2010) Adjustment coefficient for risk processes in some dependent contexts. Submitted for publication.Google Scholar
Dedecker, J., Doukhan, P., Lang, G., León, J.R., Louhichi, S. and Prieur, C. (2007) Weak Dependence: With Examples and Applications. Lecture Notes in Statistics 190, Springer-Verlag, New York.CrossRefGoogle Scholar
De Finetti, B. (1957) Su un'impostazione alternativa della teoria collecttiva del rischio. Transactions of the International Congress of Actuaries 2, 433443.Google Scholar
Dickson, D.C.M. (1994) Some comments on the compound binomial model. ASTIN Bulletin 24, 3345.CrossRefGoogle Scholar
Dickson, D.C.M. (2005) Insurance Risk and Ruin. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Freeland, R.K. (1998) Statistical analysis of discrete time series with applications to the analysis of workers compensation claims data. PhD thesis, University of British Columbia, Canada.Google Scholar
Freeland, R.K. and Mccabe, B.P.M. (2004) Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis 25(5), 701722.CrossRefGoogle Scholar
Frees, E.J. and Wang, P. (2006) Copula credibility for aggregate loss models. Insurance: Mathematics and Economics 38(2), 360373.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation for Insurance Education, Philadelphia.Google Scholar
Gerber, H.U. (1982). Ruin theory in the linear model. Insurance: Mathematics and Economics 1, 177184.Google Scholar
Gerber, H.U. (1988a) Mathematical fun with the compound binomial process. ASTIN Bulletin 18, 161168.CrossRefGoogle Scholar
Gerber, H.U. (1988b) Mathematical fun with ruin theory. Insurance: Mathematics and Economics 7, 1523.Google Scholar
Gourieroux, C. and Jasiak, J. (2004) Heterogeneous INAR(1) model with application to car insurance. Insurance: Mathematics and Economics 34, 177192.Google Scholar
Heinen, A. (2003) Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model. Munich Personal RePEc Archive.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman and Hall, London.CrossRefGoogle Scholar
Jung, R.J., Kukuk, M. and Liesenfeld, R. (2006) Time series of count data: modeling, estimation and diagnostics. Computational Statistics & Data Analysis 51(4), 23502364.CrossRefGoogle Scholar
Kedem, B. and Fokianos, K. (2002) Regression Models for Time Series Analysi. Wiley, New York.CrossRefGoogle Scholar
Kremer, E. (1995) INAR and IBNR. Blätter der DGVFM 22(2), 249253.CrossRefGoogle Scholar
Malyshkina, N.V., Mannering, F.L. and Tarko, A.P. (2009) Markov switching negative binomial models: An application to vehicle accident frequencies. Accident Analysis and Prevention 41, 217226.CrossRefGoogle ScholarPubMed
McKenzie, E. (1986) Autoregressive-moving average processes with negative binomial and geometric marginal distributions, Advances in Applied Probability 18, 679705.CrossRefGoogle Scholar
McKenzie, E. (1988) Some ARMA Models for Dependent Sequences of Poisson Counts. Advances in Applied Probability 20(4), 822835.CrossRefGoogle Scholar
McKenzie, E. (2003) Discrete variate time series. In Stochastic processes: Modelling and simulation (Shanbhag, D.N. et al., Ed.), Amsterdam: North-Holland, Handbook on Statistics, 21, 573606.CrossRefGoogle Scholar
Müller, A. and Pflug, G. (2001) Asymptotic ruin probabilities for risk processes with dependent increments. Insurance: Mathematics and Economics 28, 381392.Google Scholar
Nyrhinen, H. (1998) Rough descriptions of ruin for a general class of surplus processes. Advances in Applied Probability 30, 10081026.CrossRefGoogle Scholar
Nyrhinen, H. (1999a) On the ruin probabilities in a general economic environment. Stochastic Processes and their Applications 83, 319330.CrossRefGoogle Scholar
Nyrhinen, H. (1999b) Large deviations for the time of ruin. Journal of Applied Probability 36, 733746.CrossRefGoogle Scholar
Promislow, S.D. (1991) The probability of ruin in a process with dependent increments. Insurance: Mathematics and Economics 10, 99107.Google Scholar
Quddus, M.A. (2008) Time series count data models: An empirical application to traffic accidents. Accident Analysis and Prevention 40, 17321741.CrossRefGoogle Scholar
Shiu, E. (1989) The probability of eventual ruin in the compound binomial model. ASTIN Bulletin 19, 179190.CrossRefGoogle Scholar
Willmot, G.E. (1993) Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics 12, 133142.Google Scholar
Yang, H.L. and Zhang, L.H. (2003) Martingale method for ruin probability in an autoregressive model with constant interest rate. Probability in the Engineering and Informational Sciences 17, 183198.CrossRefGoogle Scholar
Yuen, K.C. and Guo, J.Y. (2001) Ruin probabilities for time-correlated claims in the compound binomial model. Insurance: Mathematics and Economics 29, 4757.Google Scholar
Zeger, S.L. (1988) A regression model for time series of counts. Biometrika 75(4), 621629.CrossRefGoogle Scholar
Zhang, Z., Yuen, K.C., and Li, W.K. (2007) A time-series risk model with constant interest for dependent classes of business. Insurance: Mathematics and Economics 41, 3240.Google Scholar
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