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.

The premium calculation considered in the classical bonus-malus system in automobile insurances is dependent only on a number of claims. When a policyholder declared a small claim size, he or she was penalized in the same way as those who made a large claim size. Hence, it is unfair to penalize all policyholders in like manner. In this article, we propose a model for computing the premium based on the bonus-malus system by distinguishing multiple types of claims as follows: small, middle, large, and sever claim sizes. The number of claims is considered based on discrete distributions: Poisson and binomial distributions. Besides, we introduce prior distributions, gamma, and beta distributions, conjugated with these discrete distributions. The Bayesian method is applied for updating the parameters of the model as this approach yields bonus-malus premiums. We use a real example data to demonstrate the model. The randomized neighborhood search (RNS) technique is introduced for estimating the parameters to minimize the statistical value for Akaike’s Information Criterion (AIC). This methodology allows us to obtain a good fit to the data set considered.

In this paper, we introduce a new continuous distribution mixing exponential and gamma distributions, called new Sushila distribution. We derive some properties of the distribution include: probability density function, cumulative distribution function, expected value, moments about the origin, coefficient of variation (C.V.), coefficient of skewness, coefficient of kurtosis, moment generating function, and reliability measures. The distribution includes, a special cases, the Sushila distribution as a particular case p=1/2 (θ = 1). The hazard rate function exhibits increasing. The parameter estimations as the moment estimation (ME), the maximum likelihood estimation (MLE), nonlinear least squares methods, and genetic algorithm (GA) are proposed. The application is presented to show that model to fit for waiting time and survival time data. Numerical results compare ME, MLE, weighted least squares (WLS), unweighted least squares (UWLS), and GA. The results conclude that GA method is better performance than the others for iterative methods. Although, ME is not the best estimate, ME is a fast estimate, because it is not an iterative method. Moreover, The proposed distribution has been compared with Lindley and Sushila distributions to a waiting time data set. The result shows that the proposed distribution is performing better than Lindley and Sushila distribution.