Lorenz and Polarization Orderings of the Double-Pareto Lognormal Distribution and Other Size Distributions

Polarization indices such as the Foster-Wolfson index have been developed to measure the extent of clustering in a few classes with wide gaps between them in terms of income distribution. However, Zhang and Kanbur (2001) failed to empirically find clear differences between polarization and inequality indices in the measurement of intertemporal distributional changes. This paper addresses this ‘distinction' problem on the level of the respective underlying stochastic orders, the polarization order (PO) in distributions divided into two nonoverlapping classes and the Lorenz order (LO) of inequality in distributions. More specifically, this paper investigates whether a distribution F can be either more or less polarized than a distribution H in terms of the PO if F is more unequal than H in terms of the LO. Furthermore, this paper derives conditions for the LO and PO of the double-Pareto lognormal (dPLN) distribution. The derived conditions are applicable to sensitivity analyses of inequality and polarization indices with respect to distributional changes. From this application, a suggestion for appropriate two-class polarization indices is made.

Tsallis Log-Scale-Location Models. Moments, Gini Index and Some Stochastic Orders

In this article we give theoretical results for different stochastic orders of a log-scale-location family which uses Tsallis statistics functions. These results describe the inequalities of moments or Gini index according to parameters. We also compute the mean in the case of q-Weibull and q-Gaussian distributions. The paper is aimed at analyzing the order between survival functions, Lorenz curves and (as consequences) the moments together with the Gini index (respectively a generalized Gini index). A real data application is presented in the last section. This application uses only the survival function because the stochastic order implies the order of moments. Given some supplementary conditions, we prove that the stochastic order implies the Lorenz order in the log-scale-location model and this implies the order between Gini coefficients. The application uses the estimated parameters of a Pareto distribution computed from a real data set in a log-scale-location model, by specifying the Kolmogorov–Smirnov p-value. The examples presented in this application highlight the stochastic order between four models in several cases using survival functions. As direct consequences, we highlight the inequalities between the moments and the generalized Gini coefficients by using the stochastic order and the Lorenz order.

Sufficient conditions for Lorenz ordering with common finite support

Arnold and Gokhale (2017) provided a characterization of the Lorenz inequality order between distributions with common finite support. In the more general Lorenz order context, a variety of partial orders are often used to verify the existence of Lorenz ordering. In this paper we investigate whether parallel results can be identified within the common finite support context.