Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T10:45:09.607Z Has data issue: false hasContentIssue false
  • English
  • Français

A MIXTURE MODEL FOR PAYMENTS AND PAYMENT NUMBERS IN CLAIMS RESERVING

Published online by Cambridge University Press:  17 October 2016

Patrizia Gigante ,
Liviana Picech  and
Luciano Sigalotti
Patrizia Gigante
Affiliation:
Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche ‘B. de Finetti’, University of Trieste, Piazzale Europa 1, 34127 Trieste, Italy, E-Mail: [email protected]
Liviana Picech*
Affiliation:
Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche ‘B. de Finetti’, University of Trieste, Piazzale Europa 1, 34127 Trieste, Italy
Luciano Sigalotti
Affiliation:
Dipartimento di Scienze Economiche e Statistiche, University of Udine, Via Tomadini 30, 33100 Udine, Italy, E-Mail: [email protected]
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a Tweedie's compound Poisson regression model with fixed and random effects, to describe the payment numbers and the incremental payments, jointly, in claims reserving. The parameter estimates are obtained within the framework of hierarchical generalized linear models, by applying the h-likelihood approach. Regression structures are allowed for the means and also for the dispersions. Predictions and prediction errors of the claims reserves are evaluated. Through the parameters of the distributions of the random effects, some external information (e.g. a development pattern of industry wide-data) can be incorporated into the model. A numerical example shows the impact of external data on the reserve and prediction error evaluations.

Keywords

Claims reservingTweedie's compound Poisson modelconditional mean square error of predictionhierarchical generalized linear modelsh-likelihood
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 25 - 53
Copyright
Copyright © Astin Bulletin 2016 

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

Antonio, K. and Beirlant, J. (2007) Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics, 40 (1), 58–76.Google Scholar
Antonio, K., Beirlant, J., Hoedemakers, T. and Verlaak, R. (2006) Lognormal mixed models for reported claims reserves. North American Actuarial Journal, 10 (1), 30–48.CrossRefGoogle Scholar
Booth, J.G. and Hobert, J.P. (1998) Standard errors of prediction in generalized linear mixed models. Journal of the American Statistical Association, 93, 262–272.Google Scholar
Boucher, J.P. and Davidov, D. (2011) On the importance of dispersion modeling for claims reserving: An application with the Tweedie distribution. Variance, 5 (2), 158–172.Google Scholar
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Berlin: Springer.Google Scholar
Bühlmann, H. and Moriconi, F. (2015) Credibility claims reserving with stochastic diagonal effects. ASTIN Bulletin, 45 (2), 309–353.Google Scholar
de Alba, E. (2002) Bayesian estimation of outstanding claim reserves. North American Actuarial Journal, 6 (4), 1–20.Google Scholar
England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8 (3), 443–518.Google Scholar
England, P.D. and Verrall, R.J. (2006) Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1 (2), 221–270.Google Scholar
England, P.D., Verrall, R.J. and Wüthrich, M.V. (2012). Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method. Annals of Actuarial Science, 6 (2), 258–283.Google Scholar
Gigante, P., Picech, L. and Sigalotti, L. (2013a) Claims reserving in the hierarchical generalized linear models framework. Insurance: Mathematics and Economics, 52, 381–390.Google Scholar
Gigante, P., Picech, L. and Sigalotti, L. (2013b) Prediction error for credible claims reserves: an h-likelihood approach. European Actuarial Journal, 3 (2), 453–470.Google Scholar
Gisler, A. and Wüthrich, M. (2008) Credibility for the chain ladder reserving method. ASTIN Bulletin, 38 (2), 565–600.CrossRefGoogle Scholar
Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8, 77–90.Google Scholar
Jørgensen, B. (1987) Exponential dispersion models. Journal Royal Statistical Society B, 49, 127–162.Google Scholar
Jørgensen, B. (1997) Theory of Dispersion Models. London: Chapman & Hall.Google Scholar
Jørgensen, B. and de Souza, M.C.P. (1994) Fitting Tweedie's compound Poisson model to insurance data. Scandinavian Actuarial Journal, 1994 (1), 69–93.Google Scholar
Lee, Y. and Ha, I.D. (2010) Orthodox BLUP versus h-likelihood methods for inferences about random effects in Tweedie mixed models. Statistics and Computing, 20 (3), 295–303.Google Scholar
Lee, Y. and Nelder, J.A. (1996) Hierarchical generalized linear models (with Discussion). Journal of the Royal Statistical Society B, 58, 619–678.Google Scholar
Lee, Y. and Nelder, J.A. (2001) Hierarchical generalized linear models: A synthesis of generalized linear models, random-effect models and structured dispersion. Biometrika, 88 (4), 987–1006.Google Scholar
Lee, Y., Nelder, J.A. and Pawitan, Y. (2006) Generalized Linear Models with Random Effects. Unified Analysis via h-Likelihood. Boca Raton: Chapman and Hall/CRC.Google Scholar
Mack, T. (2000) Credible claims reserves: The Benktander method. ASTIN Bulletin, 30 (2), 333–347.Google Scholar
Maiti, T., Ren, H. and Sinha, S. (2014) Prediction error of small area predictors shrinking both means and variances. Scandinavian Journal of Statistics. Theory and Applications, 41 (3), 775–790.Google Scholar
Miranda-Martinez, M.D., Nielsen, B., Nielsen, J.P. and Verrall, R.J. (2011) Cash flow simulation for a model of outstanding liabilities based on claim amounts and claim numbers. ASTIN Bulletin, 41 (1), 107–129.Google Scholar
Miranda-Martinez, M.D., Nielsen, J.P. and Verrall, R.J. (2012) Double chain ladder. ASTIN Bulletin, 42 (1), 59–76.Google Scholar
Miranda-Martinez, M.D., Nielsen, J.P. and Wüthrich, M.V. (2012) Statistical modelling and forcasting of outstanding liabilities in non-life insurance. SORT, 36 (2), 195–218.Google Scholar
Nelder, J.A. and Lee, Y. (1991). Generalised linear models for the analysis of Taguchi-type experiments. Applied Stochastic Models and Data Analysis, 7, 107–120.Google Scholar
Nelder, J.A. and Pregibon, D. (1987) An extended quasi-likelihood function. Biometrika, 74, 221–232.Google Scholar
Nelder, J.A. and Verrall, R.J. (1997) Credibility theory and generalized linear models. ASTIN Bulletin, 27 (1), 71–82.Google Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modelling of outstanding liabilities incorporating claim count uncertainty. North American Actuarial Journal, 6 (1), 113–128.Google Scholar
Ohlsson, E. (2008) Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal, 4, 301–314.Google Scholar
Ohlsson, E. and Johansson, B. (2006) Exact credibility and Tweedie models. ASTIN Bulletin, 36 (1), 121–133.Google Scholar
Saluz, A. (2015) Prediction uncertainties in the Cape Cod reserving method. Annals of Actuarial Science, 9 (2), 239–263.Google Scholar
Saluz, A., Gisler, A. and Wüthrich, M.V. (2011) Development pattern and prediction error for the stochastic Bornhuetter-Ferguson claims reserving method. ASTIN Bulletin, 41 (2), 279–313.Google Scholar
Smyth, G.K. (1989) Generalized linear models with varying dispersion. Journal of the Royal Statistical Society B, 51, 47–60.Google Scholar
Smyth, G.K. and Jørgensen, B. (2002) Fitting Tweedie's compound Poisson model to insurance claims data: Dispersion modelling. ASTIN Bulletin, 32 (1), 143–157.Google Scholar
Smyth, G.K. and Verbyla, A.P. (1999) Adjusted likelihood methods for modelling dispersion in generalized linear models. Environments, 10, 696–709.Google Scholar
Taylor, G. (2000) Loss Reserving. An Actuarial Perspective. Boston: Kluwer Academic Publishers.Google Scholar
Taylor, G. (2015) Bayesian chain ladder models. ASTIN Bulletin, 45, 75–99, doi:10.1017/asb.2014.25.Google Scholar
Verrall, R.J. (2004) A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8 (3), 67–89.Google Scholar
Verrall, R.J. and England, P.D. (2005) Incorporating expert opinion into a stochastic model for the chain-ladder technique. Insurance: Mathematics and Economics, 37 (2), 355–370.Google Scholar
Verrall, R.J., Nielsen, J.P. and Jessen, A. (2010) Including count data in claims reserving. ASTIN Bulletin, 40 (2), 871–887.Google Scholar
Wüthrich, M.V. (2003) Claims reserving using Tweedie's compound Poisson model. ASTIN Bulletin, 33 (2), 331–346.Google Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: Wiley.Google Scholar