Abstract
This article proposes new distributions for claim frequency and severity, specifically tailored for a bonus-malus system in automobile insurance. The mixed Poisson with weighted quasi Lindley distribution is recommended for modeling claim frequency, while the mixed exponential with weighted quasi Lindley distribution is suggested for modeling claim severity. To estimate insurance premiums, the Bayesian method is employed, incorporating both frequency and severity distributions. The study validates the proposed models using real data from an Australian insurance company, which includes 67,856 policies. The assessment of model adequacy indicates that the Poisson-weighted quasi Lindley distribution is a suitable fit for modeling claim frequency, while the exponential-weighted quasi Lindley distribution is appropriate for modeling claim severity. Overall, the results suggest that the proposed models offer optimal premium estimations, considering both claim frequency and severity, which can lead to fairer pricing and increased customer appeal during claim occurrences compared to conventional models.