Stochastic Dominance with Ordinal Variables: Conditions and a Test

A re-emerging literature on robustness in multidimensional welfare and poverty comparisons has revived interest in multidimensional stochastic dominance. Considering the widespread use of ordinal variables in wellbeing measurement, and particularly in composite indices, I derive multivariate stochastic dominance conditions for ordinal variables. These are the analogues of the conditions for continuous variables (e.g., Bawa, 1975 Bawa , V. S. ( 1975 ). Optimal rules for ordering uncertain prospects . Journal of Financial Economics 2 : 95 – 121 .[Crossref] , [Google Scholar], and Atkinson and Bourguignon, 1982 Atkinson , A. , Bourguignon , F. ( 1982 ). The comparison of multi-dimensioned distributions of economic status . Review of Economic Studies XLIX : 183 – 201 .[Crossref], [Web of Science ®] , [Google Scholar]). The article also derives mixed-order-of-dominance conditions for any type of variable. Then I propose an extension of Anderson's nonparametric test in order to test these conditions for ordinal variables. In addition, I propose the use of vectors and matrices of positions in order to handle multivariate, multinomial distributions. An empirical application to multidimensional wellbeing in Peru illustrates these tests.