Explicit expressions for the Leimkuhler curve in parametric families

Abstract

In this paper we obtain the Leimkuhler curve in the case of some important statistical distributions proposed in the informetrics and econometrics literature. In this way, we complete the previous work of Burrell [Burrell, Q. L. (2005). Symmetry and other transformation features of Lorenz/Leimkuhler representations of informetric data. Information Processing and Management, 41, 1317–1329], where several open problems were stated. To do this, we use a recent and general definition of the Leimkuhler curve proposed by Sarabia [Sarabia, J. M. (2008a). A general definition of the Leimkuhler curve. Journal of Informetrics, 2, 156-163], and a new representation of the Leimkuhler curve in terms of the first-moment distribution of the population. Specifically, we obtain the Leimkuhler curve of the following distributions: classical and exponentiated Pareto distributions; three-parameter lognormal distribution; generalized gamma distribution, which includes to the exponential and classical gamma distributions among others; generalized beta distribution of the first kind and generalized beta distribution of the second kind, which includes as particular or limiting cases next important families like beta distribution of the second kind, Singh–Maddala, Dagum, Fisk or Lomax distributions. All the obtained Leimkuhler curves can be computed easily.