Bayesian Assessment of Lorenz and Stochastic Dominance using a Mixture of Gamma Densities

Because of their applicability for ordering distributions within general classes of utility and social welfare functions, tests for stochastic and Lorenz dominance have attracted considerable attention in the literature. To date the focus has been on sampling theory tests, with some tests having a null hypothesis that X dominates Y (say), and others having a null hypothesis that Y does not dominate X. These tests can be performed in both directions, with X and Y reversed. We propose a Bayesian approach for assessing Lorenz and stochastic dominance where the three hypotheses (i) X dominates Y, (ii) Y dominates X, and (iii) neither Y nor X is dominant, are treated symmetrically. Posterior probabilities for each of the three hypotheses are obtained by estimating the distributions and counting the proportions of MCMC draws that satisfy each of the hypotheses. We apply the proposed approach to samples of Indonesian income distributions for 1999, 2002, 2005 and 2008. To ensure flexible modelling of the distributions, mixtures of gamma densities are fitted for each of the years. We introduce probability curves that depict the probability of dominance at each population proportion and which convey valuable information about dominance probabilities for restricted population proportions relevant when studying poverty orderings. The results are compared with those from some sampling theory tests and the probability curves are used to explain seemingly contradictory outcomes.