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A Nonhomogeneous Poisson Hidden Markov Model for Claim Counts

Published online by Cambridge University Press:  09 August 2013

Yi Lu  and
Leilei Zeng
Yi Lu
Affiliation:
Department of Statistics & Actuarial Science, Simon Fraser University, 8888 University Drive Burnaby, BC, CanadaV5A 1S6, Fax: 1.778.782.4368, E-Mail: yilu@sfu.ca
Leilei Zeng
Affiliation:
Department of Statistics & Actuarial Science, University of Waterloo, 200 University Avenue West Waterloo, Ontario, CanadaN2L 3G1, Fax: 1.519.746.1875, E-Mail: lzeng@math.uwaterloo.ca
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Abstract

We propose a nonhomogeneous Poisson hidden Markov model for a time series ofclaim counts that accounts for both seasonal variations and random fluctuations in the claims intensity. It assumes that the parameters of the intensity function for the nonhomogeneous Poisson distribution vary according to an (unobserved) underlying Markov chain. This can apply to natural phenomena that evolve in a seasonal environment. For example, hurricanes that are subject to random fluctuations (El Niño-La Niña cycles) affect insurance claims. The Expectation-Maximization (EM) algorithm is used to calculate the maximum likelihood estimators for the parameters of this dynamic Poisson hidden Markov model. Statistical applications of this model to Atlantic hurricanes and tropical storms data are discussed.

Keywords

Poisson hidden Markov modelNonhomogeneous Poisson processSeasonalityIntensity functionHurricanes and tropical stormsMaximum likelihood estimationEM algorithm
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 1 , May 2012 , pp. 181 - 202
Copyright
Copyright © International Actuarial Association 2012

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References

Albert, P.S. (1991) A two-state Markov mixture model for a time series of epileptic seizure counts. Biometrics 47, 1371–1381.Google Scholar
Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6), 716–723.Google Scholar
Altman, R.M. and Petkau, A.J. (2005) Application of hidden Markov models to multiple sclerosis lesion count data. Statistics in Medicine 24, 2335–2344.Google Scholar
Baum, L.E., Petrie, T., Soules, G. and Weiss, N. (1970) A maximization technique occurring in the statistical estimation for probabilistic functions of Markov chains. The Annals of Mathematical Statistics 41, 164–171.Google Scholar
Beard, R.E., Pentikäinen, T. and Pesonen, E. (1984) Risk Theory 3rd ed. Chapman & Hall, London.Google Scholar
Bulla, J. and Berzel, A. (2008) Computational issues in parameter estimation for stationary hidden Markov models. Comp. Stat. 23(1), 1–18.CrossRefGoogle Scholar
Cole, J.D. and Pfaff, S.R. (1997) A climatology of tropical cyclones affecting the Texas coast during El Niño/non-El Niño years: 1990-1996. Technical Attachment SR/SSD 97-37, National Weather Service Office, Corpus Christi, Texas, (http://www.srh.noaa.gov/topics/attach/html/ssd97-37.htm).Google Scholar
Cooper, B. and Lipsitch, M. (2004) The analysis of hospital infection data using hidden Markov models. Biostatistics 5(2), 223–237.CrossRefGoogle ScholarPubMed
Cox, D.R. (1981) Statistical analysis of time series: some recent developments. Scandinavian Journal of Statistics, 8(2), 93–115.Google Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries. Chapman & Hall, London.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), 701–722.Google Scholar
Freeland, R.K., Latour, A. and Oraichi, D. (2006) Integer-valued GARCH process. Journal of Time Series Analysis 27(6), 923–942.CrossRefGoogle Scholar
Garrido, J. and Lu, Y. (2004) On double periodic non-homogeneous Poisson processes. Bulletin of the Association of Swiss Actuaries 2, 195–212.Google Scholar
Gart, J.J. and Pettigrew, H.M. (1970) On the conditional moments of the k-statistics for the Poisson distribution. Biometrika 57(3), 661–664.CrossRefGoogle Scholar
Grandell, J. (1991) Aspects of Risk Theory. Springer-Verlag, Berlin.Google Scholar
Gray, W.M. (1984) Atlantic seasonal hurricane frequency. Part I: El Niño and 30 mb quasibiennial oscillation inl uences. Monthly Weather Review 112, 1649–1668.Google Scholar
Heinen, A. (2003) Modelling time series count data: an autoregressive conditional Poisson model. Center for Operations Research and Econometrics (CORE) Discussion Paper No. 2003-63, University of Louvain, Belgium.Google Scholar
Hóhle, M. and Paul, M. (2008) Count data regression charts for the monitoring of surveillance time series. Computational Statistics and Data Analysis 52, 4357–4368.CrossRefGoogle Scholar
Hughes, J.P. and Guttorp, P. (1999) A non-homogeneous hidden Markov model for precipitation occurrence. Applied Statistics 48, 15–30.Google Scholar
Juang, B.H. and Rabiner, L.R. (1991) Hidden Markov models for speech recognition. Technometrics 33, 251–272.CrossRefGoogle Scholar
Jung, R.C., Kukuk, M. and Liesenfeld, R. (2006) Time series of count data: modelling, estimation and diagnostics. Computational Statistics and Data Analysis 51, 2350–2364.Google Scholar
Katz, R.W. (1981) On some criteria for estimating the order of a Markov chain. Technometrics 23(3), 243–249.Google Scholar
Katz, R.W. (2002) Stochastic modeling of hurricane damage. Journal of Applied Meteorology 41, 754–762.Google Scholar
Landreneau, D. (2001) Atlantic tropical storms and hurricanes affecting the United States: 1899-2002. NOAA Technical Memorandum NWS SR-206 (Updated through 2002), National Weather Service Office, Lake Charles, Louisiana, (http://www.srh.noaa.gov/lch/research/tropical.htm).Google Scholar
Leroux, B.G. and Puterman, M.L. (1992) Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models. Biometrics 48, 545–558.Google Scholar
Lu, Y. and Garrido, J. (2005) Doubly periodic non-homogeneous Poisson models for hurricane data. Statistical Methodology 2, 17–35.Google Scholar
Lu, Y. and Garrido, J. (2007) Regime-switching periodic models for claim counts. North American Actuarial Journal 10, 235–248.Google Scholar
MacDonald, I.L. and Zucchini, W. (1997) Hidden Markov and Other Models for Discrete-valued Time Series. Chapman & Hall, London.Google Scholar
MacKay, R.J. (2002) Estimating the order of a hidden Markov model. The Canadian Journal of Statistics 30(4), 573–589.Google Scholar
McKenzie, E. (2003) Discrete variate time series. In Stochastic Processes: Modelling and Simulation (eds Shanbhag, D.N. and Rao, C.R.). Amsterdam: Elsevier, 573–606.Google Scholar
Paroli, R., Redaelli, G. and Spezia, L. (2000) Poisson hidden Markov models for time series of overdispersed insurance counts. ASTIN Colloquium, Porto Cervo, Italy.Google Scholar
Pielke, R.A. Jr., Gratz, J., Landsea, C.W., Collins, D., Saunders, M.A. and Musulin, R. (2008) Normalized hurricane damage in the United States: 1900-2005. Natural Hazards Review 9, 29–42.Google Scholar
Pielke, R.A. Jr., and Landsea, C.W. (1999) La Niña, El Niño, and Atlantic hurricane damages in the United States. Bull. Amer. Meteor. Soc. 80, 2027–2033.Google Scholar
Quddus, M.A. (2008) Time series count data models: an empirical application to traffic accidents. Accident Analysis and Prevention 40, 1732–1741.CrossRefGoogle ScholarPubMed
Rao, C.R. and Chakravarti, I.M. (1956) Some small sample tests of significance for a Poisson distribution. Biometrics 12, 264–282.Google Scholar
Rydén, T. (1994) Parameter estimation for Markov modulated Poisson processes. Stochastic Models 10, 795–829.Google Scholar
Schwarz, G.E. (1978) Estimating the dimension of a model. Annals of Statistics 6(2), 461–464.Google Scholar
Trenberth, K.E. (1997) The definition of El Niño. Bull. Amer. Meteor. Soc. 78, 2771–2777.Google Scholar
Weiss, C.H. (2008) Thinning operations for modelling time series of counts – a survey. Advances in Statistical Analysis 92(3), 319–341.Google Scholar
Weiss, C.H. (2009) Modelling time series of counts with overdispersion. Statistical Methods and Applications 18, 507–519.CrossRefGoogle Scholar
Wu, C.F.J. (1983) On the convergence properties of the EM algorithm. The Annals of Statistics 11, 95–103.CrossRefGoogle Scholar
Zeger, S.L. (1988) A regression model for time series of counts. Biometrika 75(4), 621–629.Google Scholar
Zucchini, W. and MacDonald, I.L. (2009) Hidden Markov Models for Time Series: An Introduction Using R, 2nd Ed. Chapman and Hall/CRC, London.Google Scholar