The bonus-malus system (BMS) is one of the most widely used tools in merit-rating automobile insurance, with the primary goal of ensuring that fair premiums are paid by all policyholders. The traditional BMS is dependent only on the claim frequency. Thus, an insured person who makes a claim with a small severity is penalized unfairly compared to an individual who makes a large severity claim. This study proposes a model for estimating the bonus-malus premium by employing a limit value (monetary unit) which distinguishes claim size into small and large based on claim frequency and claim severity distributions. This assists in determining the penalties for policyholders with claim sizes falling above and below the limit value. The number of claims is assumed to follow a Poisson distribution, and the total number of claims with a size greater than the limit value is considered a binomial distribution. The underlying risk of each policyholder is assumed to follow a beta Lindley distribution and is referred to as the prior distribution. Each policyholder’s claim size is also assumed to follow a gamma distribution, with the Lindley distribution considered as the prior distribution. Bonus-malus premiums are calculated following the Bayesian method. Practical examples using an actual data set are provided, and the results generated are compared to those produced using the traditional Poisson binomial-exponential beta model. This methodology provides a more equitable mechanism for penalizing policyholders in the portfolio.

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This article proposes a model that uses the adjusted mixture cosine model of two components with Markov chain (MC2MC) for predicting the monthly rainfall with actual data from Khon Kaen meteorological station (381201) in Khon Kaen province, Thailand. The data considers 31 years of historical data from January 1991 to December 2021. The evaluation is measured by the root mean square error (𝑅𝑀𝑆𝐸) and the 𝑅 2 values. We found that the mixture cosine model has 𝑅𝑀𝑆𝐸 and 𝑅 2 values of 70.72 and 52.49%, respectively, and the MC2MC model has 𝑅𝑀𝑆𝐸 and 𝑅 2 values of 42.43 and 82.53%, respectively.

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