Grouped data in the form of income shares have conventionally been used to estimate income inequality due to the lack of individual records. We present a systematic evaluation of the performance of parametric distributions and non-parametric techniques to estimate economic inequality using more than 3300 data sets. We also provide guidance on the choice between these two approaches and their estimation, for which we develop the GB2group R package. Our results indicate that even the simplest parametric models provide reliable estimates of inequality measures. The non-parametric approach, however, fails to represent income distributions accurately.

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modelled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabiaet al., 2016), gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecheret al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.