A Fuzzy Least-Squares Estimation of a Hybrid Log-Poisson Regression and Its Goodness of Fit for Optimal Loss Reserves in Insurance

In loss reserving, the log-Poisson regression model is a well-known stochastic model underlying the chain-ladder method which is the most used method for reserving purposes. Mack (ASTIN Bull 21(01):93–109, 1991) proved that the log-Poisson model provides the same estimates as the chain-ladder method. So in this article, our objective is to improve the log-Poisson regression model in loss reserving framework. Thereby, we prove the reliability of hybrid models in loss reserving, especially when the data contain fuzziness, for example when the claims are related to body injures (Straub and Swiss in Non-life insurance mathematics, Springer, Berlin, 1988). Thus, we estimate a hybrid generalized linear model (GLM) (log-Poisson) using the fuzzy least-squares procedures (Celmis in Fuzzy Sets Syst 22(3):245–269, 1987a; Math Model 9(9):669–690, 1987b; D’Urso and Gastaldi in Comput Stat Data Anal 34(4): 427–440, 2000; in: Advances in classification and data analysis, Springer, 2001). We develop a new goodness of fit index to compare this new model and the classical log-Poisson regression (Mack 1991). Both the classical log-Poisson model and the hybrid one are performed on a loss reserving data. According to the goodness of fit index and the mean square error prediction, we prove that the new model provide better results than the classical log-Poisson model. This comparison can be extend to any other GLM in loss reserving.