Risk Measures Associated With Automobile Insurance Claim Losses With an Underlying Probabilistic Mixture Distribution

Loss data may often exhibit features such as multimodality and skewness that render single distributions incapable of capturing all these features. Insurance data often comprise of extremely large losses, of which single distributions may inadequately capture their varying features of different sizes. In sequel, this will lead to estimation errors of some important quantities such as risk measures related to the data. In this regard, we propose the use of mixture distributions as the underlying probabilistic distribution of automobile claims data for risk estimation. It is prudent to ensure that the right probabilistic distributions are fitted to insurance data in order not to wrongly estimate associated parameters. This paper therefore, employed a two-component mixture distribution to describe automobile insurance losses from Ghana using 8916 data points. We estimated 240 mixture distributions for the automobile insurance losses. Using some goodness-of-fit criteria, the results of the top-10 mixture distributions are presented. The mixing weights for each mixture distribution is also estimated and presented. Value at risk (VaR) and tail value at risk (TVaR) were then estimated and presented for the top-10 mixture distributions at 95% and 99% security levels. A comparison of the VaR and TVaR estimates for the best mixture distribution with a nonparametric approach shows that the best mixture distribution outperformed it at the 95% percentile, while the opposite was true at the 99% percentile. However, only the Burr–Burr mixture distribution satisfied the Kupiec’s test at both confidence levels