A Flexible Geometric Mixture Distribution for Actuarial Claim Counts

This paper introduces a new two-parameter discrete distribution obtained by mixing the geometric distribution with a particular case of the Log-Gamma distribution; equivalently, it is a geometric distribution whose success probability is driven by a gamma latent variable. The proposed model is unimodal, presents a zero vertex, and accommodates over-dispersion, which are features commonly observed in automobile insurance-claim data. For α<1, it is shown to be more dispersed and to have more zeros than the geometric distribution with the same mean. Closed-form expressions are derived for several actuarial quantities and reliability measures, including the failure rate, mean residual life, Value at Risk (VaR), through the Lambert W function, and Tail Value at Risk (TVaR). In addition, the geometric distribution is obtained as a limiting case of the proposed model when α<1. We study parameter estimation using a zero-proportion and mean estimator, maximum-likelihood estimation, and an ECME algorithm. The practical performance of the proposed model is illustrated through a Monte Carlo study that explicitly accounts for boundary solutions, and through three automobile insurance datasets, where it is compared with the Poisson and negative binomial distributions using asymptotic and parametric bootstrap goodness-of-fit tests. The proposed model performs as well as the negative binomial distribution in two classical datasets, whereas for a portfolio of 100,000 policies it attains the smallest AIC but, like the negative binomial model, it is rejected by the goodness-of-fit tests.