Actuarial measures, aggregate loss models, and insurance applications of the sine exponentiated Burr XII distribution

The appropriate use of trigonometric functions in distribution theory can not be over-emphasized. In this paper, an extension of the exponentiated Burr XII distribution is proposed via the sine-Gfamily of distributions. This distribution is known as the sine exponentiated Burr XII distribution. The density is decreasing and right skewed with varying degree of peakedness whereas the hazard rate is decreasing and unimodal shaped. Mathematical properties such as the quantile function, moments, incomplete moments, moment generating function, and order statistics are derived. Also, actuarial measures such as value at risk, tail value at risk, and tail variance are derived and studied. The results show that the actuarial measures of the SEBXII distribution increase with increasing confidence level an indication that the SEBXII distribution is heavy-tailed. Under collective risk model, the mean and variance of the aggregate loss distribution where the frequency distribution for the claim count is Poisson are derived and studied. Estimates of the expected payout and the variance of the payout for the aggregate loss are obtained for varying parameter values. Five estimation approaches are used namely the maximum likelihood, least squares, weighted least squares, percentile, and Anderson-Darling are employed in examining the behavior of the proposed distribution. Simulations are carried for each of the estimation method. The results show that in most of the estimation methods, the AB and MSE approaches zero as the sample size increases, an indication that the estimators are consistent. That is the estimators of the proposed distribution are consistent for all the estimation methods. The usefulness of the proposed distribution is demonstrated with two insurance loss datasets. The results show that the proposed distribution provide the best parametric fit for the two datasets. The proposed distribution is flexible and capable of modeling heavy-tailed datasets.