Weak orderings for intersecting Lorenz curves

The Lorenz dominance is a primary tool for comparison of non-negative distributions in terms of inequality. However, in most of cases Lorenz curves intersect and the ordering is not fulfilled, so that some alternative (weaker) criteria need to be to introduced. In this context, the second-degree Lorenz dominance, which emphasizes the role of the left (or right) tail of the distribution, is especially suitable for ranking single-crossing Lorenz curves. We introduce a new ordering, namely disparity dominance, which emphasizes inequality in both of the tails, and we show that, in turn, it is especially suitable for ranking double-crossing Lorenz curves. We argue that the two approaches are basically complementary, although in both cases the Gini coefficient is crucial for the ranking. Moreover, we can use some well-known results of majorization theory to obtain classes of functionals that are consistent with the aforementioned weak preorders, and that can therefore be used as finer inequality indices.