A comparative analysis of several multivariate zero-inflated and zero-modified models with applications in insurance

Claim frequency data in insurance records the number of claims on insurance policies during a finite period of time. Given that insurance companies operate with multiple lines of insurance business where the claim frequencies on different lines of business are often correlated, multivariate count modeling with dependence for claim frequency is therefore essential. Due in part to the operation of bonus-malus systems, claims data in automobile insurance are often characterized by an excess of common zeros. This feature is referred to as multivariate zero-inflation. In this paper, we establish two ways of dealing with this feature. The first is to use a multivariate zero-inflated model, where we artificially augment the probability of common zeros based on standard multivariate count distributions. The other is to apply a multivariate zero-modified model, which deals with the common zeros and the number of claims incurred in each line, given that at least one claim occurs separately. A comprehensive comparative analysis of several models under these two frameworks is conducted using the data of an automobile insurance portfolio from a major insurance company in Spain. A less common situation in insurance is the absence of some common zeros resulting from incomplete records. This feature of these data is known as multivariate zero-deflation. In this case, our proposed multivariate zero-modified model still works, as shown by the second empirical study.