The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions

A family of probability densities, which has proved useful in modelling the size distributions of various phenomena, including incomes and earnings, human settlement sizes, oil-field volumes and particle sizes, is introduced. The distribution, named herein as the double Pareto-lognormal or dPlN distribution, arises as that of the state of a geometric Brownian motion (GBM), with lognormally distributed initial state, after an exponentially distributed length of time (or equivalently as the distribution of the killed state of such a GBM with constant killing rate). A number of phenomena can be viewed as resulting from such a process (e.g. incomes, settlement sizes), which explains the good fit. Properties of the distribution are derived and estimation methods discussed. The distribution exhibits Paretian (power-law) behaviour in both tails, and when plotted on logarithmic axes, its density exhibits hyperbolic-type behaviour.