Academic journal article Journal of Risk and Insurance

Number of Accidents or Number of Claims? an Approach with Zero-Inflated Poisson Models for Panel Data

Academic journal article Journal of Risk and Insurance

Number of Accidents or Number of Claims? an Approach with Zero-Inflated Poisson Models for Panel Data

Article excerpt

ABSTRACT

The hunger for bonus is a well-known phenomenon in insurance, meaning that the insured does not report all of his accidents to save bonus on his next year's premium. In this article, we assume that the number of accidents is based on a Poisson distribution but that the number of claims is generated by censorship of this Poisson distribution. Then, we present new models for panel count data based on the zero-inflated Poisson distribution. From the claims distributions, we propose an approximation of the accident distribution, which can provide insight into the behavior of insureds. A numerical illustration based on the reported claims of a Spanish insurance company is included to support this discussion.

INTRODUCTION

In various applications involving count data, the data exhibit a high number of zero values. This led to the idea that a distribution with excess zeros can provide a good fit, such as the zero-inflated distribution. In insurance, the hunger for bonus (Philipson, 1960; Lemaire, 1977) is a well-known phenomenon that represents the fact that insureds do not report all their accidents to save bonus on the following year's premium. However, actuaries and researchers continue to model the number of claims with standard count distributions, neglecting the bonus hunger phenomenon. In this article, we assume that the number of accidents is based on a Poisson distribution but that the number of claims is generated by censorship of this Poisson distribution.

Risk classification techniques for claim counts have been the topic of many papers in the actuarial literature. Denuit et al. (2007) provides an exhaustive overview of count data models for insurance claims. For cross-sectional data, Boucher, Denuit, and Guillen (2007) studied zero-inflated models in motor insurance-claim counts and compared them to hurdle models. Boucher, Denuit, and Guillen (2008) worked on risk classification models for the number of claims, in the context of panel data but only studied classical count data models, like Poisson and negative binomial. In this article, we propose new zero-inflated models that generalize the distribution for longitudinal data by introducing random effects in the model.

We found that the generalizations of the zero-inflated Poisson distribution has interesting applications for insurance data, where the number of accidents can be compared to the number of claims. In "Number of Claims versus Number of Accidents" section, we review the standard techniques used to consider the differences between the number of claims and the number of accidents. An overview of the zero-inflated model and the basis for the construction of panel data distributions are given in the next section. In the "Multivariate Zero-Inflated Models" section, we propose various types of parameterizations of zero-inflated models for panel data. Random effects are added to the zero-inflated term and the count distribution. Another kind of multivariate distribution, comprising a special degenerated random effects distribution, is shown to have interesting properties.

In the "Predictive Distributions" section, we show that predictive distributions and their expected predictive value can be expressed in a closed form for all proposed distributions. The "Numerical Application" section contains a numerical illustration performed on the reported claims in a sample from a Spanish insurance company to support the discussion that allows us to analyze the hunger for bonus situation. We show that the expected value and the variance can differ significantly according to the model. Statistical tests to compare models are explained and a Vuong-Golden test is used to compare the fitting of nonnested models. Our results show that some of the models presented here with insurance data have a better fit than the commonly used Poisson distribution with gamma random effects.

NUMBER OF CLAIMS VERSUS NUMBER OF ACCIDENTS

In most of the bonus-malus schemes throughout the world (see Lemaire, 1995, for an overview), a reported claim causes an increase in the premium for the following years. …

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