A unit Weibull loss distribution with quantile regression and practical applications to actuarial science

A new bounded distribution called the unit Weibull loss distribution has been derived. The corresponding probability density function plots reveal that it can be used to analyze data that exhibit right skewness, left skewness, and approximately symmetric and decreasing shapes. Furthermore, the corresponding hazard rate function plots indicate that it is adequate to fit data that have J, bathtub, and modified bathtub hazard rate shapes. This makes the new distribution suitable for modeling data with complex characteristics. Statistical properties such as the quantile, moments, and moment-generating function are determined. Risk measures, including value-at-risk, tail value-at-risk, and tail variance are also calculated. Furthermore, different principles are derived for the computation of insurance premiums. The parameters of the distribution are estimated using different methods, and their performance is assessed via Monte Carlo simulations. The accuracy of the estimates is thus empirically demonstrated. A quantile regression model with responses following the unit distribution is developed. Applications of the proposed distribution and its corresponding regression model to three insurance data sets are carried out, with their performance compared with other models. The results show that they outperform the other competing models. Thus, the new methodology can serve as an alternative modeling toolkit for modeling insurance data.