A Generalized Linear Model Considering Nonlinear Effects And Its Application In The Claim Frequency Of Auto Insurance

In 2020,my country carried out the third auto insurance rate reform.Based on the pricing power of auto insurance being led by the market,it further requires the determination of auto insurance rates to be more scientific and reasonable.Therefore,the research on the determination of auto insurance rates is still of great practical significance for today’s 2021.As the claim frequency is a very important element in the determination of the rate,it is also necessary to conduct a model fitting study on it.As we all know,the generalized linear model is a classic car insurance rate determination model.As for the claim frequency,assuming it obeys the exponential distribution,establishing the relationship between the linear part of the explanatory variable and the mean value of the claim frequency transformed by the link function has its advantages but also its drawbacks.On the one hand,the distribution range is small.On the other hand,it does not include the non-linear effects of explanatory variables.For example,the explanatory variables of the generalized linear model are only a combination of linear additions,and it is difficult to fully reflect its nonlinear impact on the claim frequency.This article constructs a claim frequency model for research under the assumption of extended distribution and considering the non-linear effect information of explanatory variables and their extraction methods on the basis of the generalized linear model.Due to the limitation of the method of extracting information from the explanatory variables of the generalized linear model,it can only reflect the linear effect but difficult to express the non-linear effect,we introduce the generalized additive model.The generalized additive model adds a smoothing function to the generalized linear model.It can be in any form and has no limitation.It greatly expands the way of expressing the non-linear information of the explanatory variable relative to the claim frequency.However,the generalized additive model still has its limitations.It is the same as the generalized linear model,the distribution is still limited to the exponential family distribution,and only the model is fitted to the mean of the response variable.So we introduced the GAMLSS model,which greatly expands the range of available distributions,and simultaneously fits the mean,scale,and shape parameters of the distribution,which greatly improves the flexibility of the model.The first part of this article sorts out the research process on the claim frequency,as well as the historical research process of the generalized linear model,generalized additive model,and GAMLSS model.The second part introduces their theoretical structure,parameter estimation,model testing,and their respective advantages and disadvantages lay the theoretical foundation for empirical research.At the same time,the second part also introduces the key theoretical basis of distribution expansion—zero inflation model.The model construction idea of this paper is based on the generalized linear model,and the traditional generalized linear model and the generalized linear model considering the nonlinear effect are used to study the construction model of the claim frequency.The third part of this article introduces the construction of the claim frequency prediction model,including the construction of the claim frequency model based on GLM and GAM.The models are all carried out through three parts:theoretical structure construction,distribution selection and parameter estimation.In addition,building a model based on GAM has a very important part—the choice of non-linear effect information extraction methods.This is also the essential difference between models that consider non-linear effects and traditional generalized linear models.The third part of this article then introduces the principles of the three types of information criteria for model evaluation,which lays a theoretical foundation for the model comparison in the empirical research part of this article.The fourth part of this article-the empirical research part is the key part.On the basis of the theoretical parts,the Australian third party loss auto insurance data is used to construct the claim frequency model.The empirical research is mainly realized through the gamlss function in the GAMLSS package of the R language.According to the expansion of the distribution and the method of extracting nonlinear information,the claim frequency model is constructed,and through the comparison and analysis of the three types of information criteria,we find that the generalized linear model that takes into account the nonlinear effect is indeed better than the traditional generalized linear model,and according to empirical research It is found that under the assumption of Poisson inverse Gaussian distribution,the nonlinear information extraction method of P-spline(SBC algorithm)has the best fitting effect.The fifth part of the article analyzes the empirical results,namely,Poisson inverse Gaussian distribution and P-spline(SBC algorithm)information extraction methods have the best model fitting effect because: The hypothetical distribution of the claim frequency is closer to reality.The nonlinear effect and the information extraction method is more suitable for the impact of real explanatory variables on the claim frequency.Moreover,we found that under the Bayesian information criterion,the best way to extract information about nonlinear effects in different distributions is often P-spline(SBC algorithm).We come to a conclusion that generalized linear models that consider nonlinear effects tend to have better fitting effects and greater flexibility and maneuverability than traditional generalized linear models.In addition,when selecting the information extraction method of non-linear effects,the purpose of the initial selection method should be considered,whether it is for the model to have a higher degree of fit,or to seek a balance between the degree of model fit and complexity.It gives us a clearer direction for our thinking of constructing a claim frequency model in practice.Taking into account the nonlinear effect improves the fit between the claim number model and the real claim,making the rate determination more scientific,reasonable and realistic.