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, the author derives multivariate stochastic dominance conditions for ordinal variables. These are the analogues of the conditions for continuous variables (e.g., Bawa, 1975, and Atkinson and Bourguignon, 1982). The article also derives mixed-order-of-dominance conditions for any type of variable. Then the author proposes an extension of Anderson's nonparametric test in order to test these conditions for ordinal variables. In addition, he 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.
Yalonetzky, G. Stochastic Dominance with Ordinal Variables: Conditions and a Test. Econometric Reviews (2012) 32 (1) 126-163. [DOI: 10.1080/07474938.2012.690653]