Summary: The use of partial orders has been popularized as a way to conduct social evaluations using only minimal normative assumptions. Generically, this process involves comparing continuously indexed curves that are uniquely determined by the cumulative distributions of the individual attributes under study. In the literature on income poverty and inequality, for example, pairwise comparisons of entire income distributions and their respective Lorenz curves are routinely performed in order to characterize rankings of poverty, inequality, and welfare. In this article, we focus on the inferential problem that arises whenever such comparisons are made in the absence of census data. Statistical inference in these situations is particularly complex due to the fact that comparing curves invariably gives rise to four possibilities: the true population curves are equal, the first curve lies below the second, the second lies below the first, or the curves cross. To address this four-decision problem, we introduce a two-stage test that has good power and fine control over misclassification error rates.