The generalized linear model (GLM) is a statistical model which has been widely used in actuarial practices, especially for insurance ratemaking. Due to the inherent longitudinality of property and casualty insurance claim datasets, there have been some trials of incorporating unobserved heterogeneity of each policyholder from the repeated observations. To achieve this goal, random effects models have been proposed, but theoretical discussions of the methods to test the presence of random effects in GLM framework are still scarce. In this article, the concept of Bregman divergence is explored, which has some good properties for statistical modeling and can be connected to diverse model selection diagnostics as in Goh and Dey [(2014) Journal of Multivariate Analysis, 124, 371–383]. We can apply model diagnostics derived from the Bregman divergence for testing robustness of a chosen prior by the modeler to possible misspecification of prior distribution both on the naive model, which assumes that random effects follow a point mass distribution as its prior distribution, and the proposed model, which assumes a continuous prior density of random effects. This approach provides insurance companies a concrete framework for testing the presence of nonconstant random effects in both claim frequency and severity and furthermore appropriate hierarchical model which can explain both observed and unobserved heterogeneity of the policyholders for insurance ratemaking. Both models are calibrated using a claim dataset from the Wisconsin Local Government Property Insurance Fund which includes both observed claim counts and amounts from a portfolio of policyholders.

广义线性模型（GLM）是一种统计模型，已广泛用于精算实践中，尤其是在制定保险费率时。由于财产和伤亡保险理赔数据集具有固有的纵向性，因此已经进行了一些试验，将反复观察得出的每个投保人的未观察到的异质性纳入其中。为了实现这个目标，已经提出了随机效应模型，但是对于在GLM框架中测试随机效应的存在方法的理论讨论仍然很少。本文探讨了Bregman散度的概念，该概念具有一些良好的统计建模属性，可以与Goh和Dey [（2014）Multivariate Analysis，124，371–383]。我们可以将源自Bregman散度的模型诊断应用于测试建模者选择的先验的鲁棒性，以评估朴素模型上先验分布的可能错误指定，前提是假设随机效应遵循点质量分布作为先验分布，并且提出模型，假设随机效应具有连续的先验密度。这种方法为保险公司提供了一个具体的框架，用于测试索赔频率和严重性中是否存在非恒定随机效应，此外，还可以提供适当的层次模型，该模型可以解释保险费率制定过程中被观察者和未观察到的异质性。