Joint Partially Models for Dependent Frequency and Severity of Insurance Claims with Using Spline Functions

In this paper, the claim counts and amounts without and with missing values are assumed to be correlated in non-life insurance. The approach consists of fitting joint random effects partially models to the factorization of the joint distribution of the claim counts and the average claim amount. Also, the joint partially aggregate claims models for correlated claim count and amounts responses, without and with missing values in both variables, and their nonignorable missing mechanisms are proposed. We consider the use of different spline functions to explore the non-linear effects of time on the joint partially aggregate claims model. A full likelihood-based approach that yields maximum likelihood estimates of the joint partially models parameters for correlated claim count and amounts responses is used. For data with nonignorable missing responses in both variables, a pure premium formula is also suggested. A comparison between the pure premium obtained by the independent aggregate claims model and the resulting pure premium obtained by the joint partially aggregate claims models is also given by a simulation study. A common