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  • Zarina Nukeshtayeva Oflaz (a1) (a2), Ceylan Yozgatligil (a3) and A. Sevtap Selcuk-Kestel (a4)


In this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.


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Ambagaspitiya, R.S. (1999) On the distributions of two classes of correlated aggregate claims. Insurance: Mathematics and Economics, 24(3), 301308.
Äuerle, N. and Müller, A. (1998) Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bulletin: The Journal of the IAA, 28(1), 5976.
Avanzi, B., Wong, B. and Yang, X. (2016) A micro-level claim count model with overdispersion and reporting delays. Insurance: Mathematics and Economics, 71, 114.
Badescu, A.L., Lin, X.S. and Tang, D. (2016) A marked Cox model for the number of IBNR claims: Theory. Insurance: Mathematics and Economics, 69, 2937.
Boudreault, M., Cossette, H. and Marceau, Ã. (2014) Risk models with dependence between claim occurrences and severities for Atlantic hurricanes. Insurance: Mathematics and Economics, 54, 123132.
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics. Shaumburg, IL: The Society of Actuaries.
Cheung, L.W.K. (2004) Use of runs statistics for pattern recognition in genomic DNA sequences. Journal of Computational Biology, 11(1), 107124.
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1), 138.
Denuit, M., Lefèvre, C. and Utev, S. (2002) Measuring the impact of dependence between claims occurrences. Insurance: Mathematics and Economics, 30(1), 119.
Dhaene, J. and Goovaerts, M.J. (1997) On the dependency of risks in the individual life model. Insurance: Mathematics and Economics, 19(3), 243253.
Dias, J.G. and Ramos, S.B. (2014) Dynamic clustering of energy markets: An extended hidden Markov approach. Expert Systems with Applications, 41(17), 77227729.
Forney, G.D. (1973) The Viterbi algorithm. Proceedings of the IEEE, 61(3), 268278.
Garanti Securities (2016) Turkish Non-life Insurance Sector - Waiting for the Dust to Settle. Istanbul: Garanti Securities.
Garrido, J., Genest, C. and Schulz, J. (2016) Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Mathematics and Economics, 70, 205215.
Hassan, M.R. and Nath, B. (2005) Stock market forecasting using hidden Markov model: A new approach. Proceedings of the 5th International Conference on Intelligent Systems Design and Applications, 2005, ISDA’05, pp. 192196. IEEE, Warshaw, Poland.
Hudecová, Š. and Pešta, M. (2013) Modeling dependencies in claims reserving with GEE. Insurance: Mathematics and Economics, 53(3), 786794.
Inflation rates (2016),, last accessed 13 July 2016.
JLT Sigorta ve Reasurans Brokerligi A.S. (2017) Turkish Insurance Market Outlook 2016-17. Istanbul: JLT Sigorta ve Reasurans Brokerligi A.S.
Krogh, A., Brown, M., Mian, I.S., Sjölander, K. and Haussler, D. (1994) Hidden Markov models in computational biology: Applications to protein modeling. Journal of molecular biology, 235(5), 15011531.
Little, R.J. and Rubin, D.B. (2014) Statistical Analysis with Missing Data. Hoboken: John Wiley & Sons.
Oflaz, Z. (2016) Bivariate Hidden Markov model to capture the claim dependency. Unpublished M.Sc. Thesis, METU, Department of Statistics.
Paroli, R., Redaelli, G. and Spezia, L. (2000) Poisson hidden Markov models for time series of overdispersed insurance counts. Proceedings of the XXXI International ASTIN Colloquium (Porto Cervo, 18–21 September 2000). Roma: Instituto Italiano degli Attuari.
Pešta, M. and Okhrin, O. (2014) Conditional least squares and copulae in claims reserving for a single line of business. Insurance: Mathematics and Economics, 56, 2837.
Pinquet, J. (2000) Experience rating through heterogeneous models. In Handbook of Insurance, pp. 459500. Netherlands: Springer.
Rabiner, L.R. (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257286.
Rabiner, L.R. and Juang, B.H. (1993) Fundamentals of Speech Recognition, Vol. 14. Englewood Cliffs: PTR Prentice Hall.
Ren, J. (2012) A multivariate aggregate loss model. Insurance: Mathematics and Economics, 51(2), 402408.
Rocca, G. (2016) Insurance Objectives, studio D,, last accessed 12 July 2016.
Tse, Y.K. (2009) Nonlife Actuarial Models. Theory, Methods and Evaluation. New York: Cambridge University Press.
Viterbi, A. (1967) Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13(2), 260269.
Wang, G. and Yuen, K.C. (2005) On a correlated aggregate claims model with thinning-dependence structure. Insurance: Mathematics and Economics, 36(3), 456468.
Yuen, K.C., Guo, J. and Wu, X. (2002) On a correlated aggregate claims model with Poisson and Erlang risk processes. Insurance: Mathematics and Economics, 31(2), 205214.
Zucchini, W. and Guttorp, P. (1991) A hidden Markov model for space -time precipitation. Water Resources Research, 27(8), 19171923.
Zucchini, W. and MacDonald, I. L. (2009) Hidden Markov Models for Time Series: An Introduction Using R. Boca Raton: Chapman and Hall, CRC.



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