# Bayesian Assessment of Lorenz and Stochastic Dominance in Income Distributions

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## Abstract

Hypothesis tests for dominance in income distributions has received considerable attention in recent literature. See, for example, Barrett and Donald (2003), Davidson and Duclos (2000) and references therein. Such tests are useful for assessing progress towards eliminating poverty and for evaluating the effectiveness of various policy initiatives directed towards welfare improvement. To date the focus in the literature has been on sampling theory tests. Such tests can be set up in various ways, with dominance as the null or alternative hypothesis, and with dominance in either direction (X dominates Y or Y dominates X). The result of a test is expressed as rejection of, or failure to reject, a null hypothesis. In this paper we develop and apply Bayesian methods of inference to problems of Lorenz and stochastic dominance. The result from a comparison of two income distributions is reported in terms of the posterior probabilities for each of the three possible outcomes: (a) X dominates Y, (b) Y dominates X, and (c) neither X nor Y is dominant. Reporting results about uncertain outcomes in terms of probabilities has the advantage of being more informative than a simple reject / do-not-reject outcome. Whether a probability is sufficiently high or low for a policy maker to take a particular action is then a decision for that policy maker. The methodology is applied to data for Canada from the Family Expenditure Survey for the years 1978 and 1986. We assess the likelihood of dominance from one time period to the next. Two alternative assumptions are made about the income distributions –Dagum and Singh-Maddala – and in each case the posterior probability of dominance is given by the proportion of times a relevant parameter inequality is satisfied by the posterior observations generated by Markov chain Monte Carlo.

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