Modeling actuarial data with a composite lognormal-Pareto model

Abstract

The actuarial and insurance industries frequently use the lognormal and the Pareto distributions to model their payments data. These types of payment data are typically very highly positively skewed. Pareto model with a longer and thicker upper tail is used to model the larger loss data, while the larger data with lower frequencies as well as smaller data with higher frequencies are usually modeled by the lognormal distribution. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to its monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, we were motivated to look for a new avenue to remedy the situation. Here we introduce a two-parameter smooth continuous composite lognormal-Pareto model that is a two-parameter lognormal density up to an unknown threshold value and a two-parameter Pareto density for the remainder. The resulting two-parameter smooth density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation techniques and properties of this new composite lognormal-Pareto model are discussed and we compare its performance with the other commonly used models. A simulated example and a well-known fire insurance data set are analyzed to show the importance and applicability of this newly proposed composite lognormal-Pareto model.