The Beta Odd Pareto Distribution With Application to Heavy-Tailed Insurance Loss Data

We propose and study the five-parameter beta odd Pareto (BOP) distribution, obtained by applying the beta generator to the odd Pareto baseline. The construction is standard, but it leads to new results on asymptotic tail behavior and identifiability, together with applications to risk management. The density can be positively or negatively skewed, symmetric, unimodal or bathtub-shaped, and the hazard rate can be increasing, upside-down bathtub or bathtub-shaped. We prove every result that admits a closed-form derivation: an exact infinite mixture representation, the boundary behaviour of the density, regular variation of the survival function with tail index α=λβδ and closed integral forms for the quantile function, moments, mean residual life, order statistics and the maximum-likelihood score equations. A Monte Carlo study shows that maximum likelihood recovers risk-relevant functionals (median, high quantiles and value at risk) accurately, while the individual shape parameters remain weakly identified, with undercovering Wald intervals that a Bayesian alternative substantially improves. On automobile insurance claims, BOP outperforms the Pareto distribution and five extensions on likelihood- and distribution-based fit criteria and gives the most accurate value at risk and conditional tail expectation backtests at the 1% level. On a second, independent dataset (Norwegian fire claims, n=4,745), BOP again decisively outperforms its own nested submodels in and out of sample and attains the smallest distribution-based fit statistics of all seven models compared, a robustness check we report in full.