Skip to main content Accessibility help


  • Patrizia Gigante (a1), Liviana Picech (a2) and Luciano Sigalotti (a3)


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.


Corresponding author


Hide All
Antonio, K. and Beirlant, J. (2007) Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics, 40 (1), 5876.
Antonio, K., Beirlant, J., Hoedemakers, T. and Verlaak, R. (2006) Lognormal mixed models for reported claims reserves. North American Actuarial Journal, 10 (1), 3048.
Booth, J.G. and Hobert, J.P. (1998) Standard errors of prediction in generalized linear mixed models. Journal of the American Statistical Association, 93, 262272.
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), 158172.
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Berlin: Springer.
Bühlmann, H. and Moriconi, F. (2015) Credibility claims reserving with stochastic diagonal effects. ASTIN Bulletin, 45 (2), 309353.
de Alba, E. (2002) Bayesian estimation of outstanding claim reserves. North American Actuarial Journal, 6 (4), 120.
England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8 (3), 443518.
England, P.D. and Verrall, R.J. (2006) Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1 (2), 221270.
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), 258283.
Gigante, P., Picech, L. and Sigalotti, L. (2013a) Claims reserving in the hierarchical generalized linear models framework. Insurance: Mathematics and Economics, 52, 381390.
Gigante, P., Picech, L. and Sigalotti, L. (2013b) Prediction error for credible claims reserves: an h-likelihood approach. European Actuarial Journal, 3 (2), 453470.
Gisler, A. and Wüthrich, M. (2008) Credibility for the chain ladder reserving method. ASTIN Bulletin, 38 (2), 565600.
Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8, 7790.
Jørgensen, B. (1987) Exponential dispersion models. Journal Royal Statistical Society B, 49, 127162.
Jørgensen, B. (1997) Theory of Dispersion Models. London: Chapman & Hall.
Jørgensen, B. and de Souza, M.C.P. (1994) Fitting Tweedie's compound Poisson model to insurance data. Scandinavian Actuarial Journal, 1994 (1), 6993.
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), 295303.
Lee, Y. and Nelder, J.A. (1996) Hierarchical generalized linear models (with Discussion). Journal of the Royal Statistical Society B, 58, 619678.
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), 9871006.
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.
Mack, T. (2000) Credible claims reserves: The Benktander method. ASTIN Bulletin, 30 (2), 333347.
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), 775790.
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), 107129.
Miranda-Martinez, M.D., Nielsen, J.P. and Verrall, R.J. (2012) Double chain ladder. ASTIN Bulletin, 42 (1), 5976.
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), 195218.
Nelder, J.A. and Lee, Y. (1991). Generalised linear models for the analysis of Taguchi-type experiments. Applied Stochastic Models and Data Analysis, 7, 107120.
Nelder, J.A. and Pregibon, D. (1987) An extended quasi-likelihood function. Biometrika, 74, 221232.
Nelder, J.A. and Verrall, R.J. (1997) Credibility theory and generalized linear models. ASTIN Bulletin, 27 (1), 7182.
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modelling of outstanding liabilities incorporating claim count uncertainty. North American Actuarial Journal, 6 (1), 113128.
Ohlsson, E. (2008) Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal, 4, 301314.
Ohlsson, E. and Johansson, B. (2006) Exact credibility and Tweedie models. ASTIN Bulletin, 36 (1), 121133.
Saluz, A. (2015) Prediction uncertainties in the Cape Cod reserving method. Annals of Actuarial Science, 9 (2), 239263.
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), 279313.
Smyth, G.K. (1989) Generalized linear models with varying dispersion. Journal of the Royal Statistical Society B, 51, 4760.
Smyth, G.K. and Jørgensen, B. (2002) Fitting Tweedie's compound Poisson model to insurance claims data: Dispersion modelling. ASTIN Bulletin, 32 (1), 143157.
Smyth, G.K. and Verbyla, A.P. (1999) Adjusted likelihood methods for modelling dispersion in generalized linear models. Environments, 10, 696709.
Taylor, G. (2000) Loss Reserving. An Actuarial Perspective. Boston: Kluwer Academic Publishers.
Taylor, G. (2015) Bayesian chain ladder models. ASTIN Bulletin, 45, 7599, doi:10.1017/asb.2014.25.
Verrall, R.J. (2004) A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8 (3), 6789.
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), 355370.
Verrall, R.J., Nielsen, J.P. and Jessen, A. (2010) Including count data in claims reserving. ASTIN Bulletin, 40 (2), 871887.
Wüthrich, M.V. (2003) Claims reserving using Tweedie's compound Poisson model. ASTIN Bulletin, 33 (2), 331346.
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: Wiley.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed