Testing the concavity of an ordinaldominance curve

We study the asymptotic properties of a class of statistics used for testing the null hypothesis that an ordinal dominance curve is concave. The statistics are based on the Lp-distance between the empirical ordinal dominance curve and its least concave majo- rant, with 1 ≤ p ≤ ∞. We formally establish the limit distribution of the statistics when the true ordinal dominance curve is concave. Further, we establish that, when 1 ≤ p ≤ 2, the limit distribution is stochastically largest when the true ordinal dominance curve is the 45-degree line. When p > 2, this is not the case, and in fact the limit distribution diverges to infinity along a suitably chosen sequence of concave ordinal dominance curves. Our results serve to clarify, extend and amend assertions appearing previously in the literature for the cases p = 1 and p = ∞.