On the Poisson-transmuted exponential distribution and its application to frequency of claim in actuarial science

Abstract

This study proposes a new discrete distribution in the mixed Poisson paradigm to obtain a distribution that provides a better fit to skewed and dispersed count observation with excess zero. The cubic transmutation map is used to extend the exponential distribution, and the obtained continuous distribution is assumed for the parameter of the Poisson distribution. Various moment-based properties of the new distribution are obtained. The Nelder-Mead algorithm provides the fastest convergence iteration under the maximum likelihood estimation technique. The shapes of the proposed new discrete distribution are similar to those of the mixing distribution. Frequencies of insurance claims from different countries are used to assess the performance of the new proposition (and its zero-inflated form). Results show that the new distribution outperforms other competing ones in most cases. It is also revealed that the natural form of the new distribution outperforms its zeroinflated version in many cases despite having observations with excess zero counts.