NONPARAMETRIC TESTS OF DENSITY RATIO ORDERING

Abstract

We study a family of nonparametric tests of density ratio ordering between two continuous probability distributions on the real line. Density ratio ordering is satisfied when the two distributions admit a nonincreasing density ratio. Equivalently, density ratio ordering is satisfied when the ordinal dominance curve associated with the two distributions is concave. To test this property, we consider statistics based on the Lp-distance between an empirical ordinal dominance curve and its least concave majorant. We derive the limit distribution of these statistics when density ratio ordering is satisfied. Further, we establish that, when 1 ≤ p ≤ 2, the limit distribution is stochastically largest when the two distributions are equal. When 2 < p ≤ ∞, 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 = ∞. We provide numerical evidence confirming their relevance in finite samples.