When is an Exponentiated Pareto distribution infinitely divisible?

Abstract

We investigate the conditions of infinite divisibility of the Exponentiated Pareto (EP) distribution supported on the entire positive half-line $(0,\infty )$, along with its discrete analogue defined on the set of non-negative integers. The EP distribution is defined via the cumulative distribution function (CDF) ${\left[F\left(x\right)\right]}^{\alpha }$ , where $\alpha >0$ and $F\left(x\right)=1-{(1+x)}^{-\lambda },x>0$, is the CDF of Pareto Type II (Lomax) distribution with tail parameter $\lambda >0.$ The discrete counterpart is defined as the integer part of a random variable $X$ following the EP distribution. The main results assert that both the continuous and discrete versions of the EP distribution are infinitely divisible if $\alpha \in (0,1].$ A brief discussion of the Lévy process corresponding to the infinitely divisible case for $\alpha \in (0,1]$ is provided along with a real world data illustration.