A relative dissimilarity ordering in the space of distribution functions, with statistical applications

The majorization relation express the fact that a given vector y is “more dissimilar” to a vector of constants than another vector x. An extended version of this relation can be applied to random variables X and Y with equal means and can be interpreted by saying that Y is “more variable than” X. In this paper we show that the majorization ordering (in a sort of weighted version) may used to define a relative dissimilarity ordering between distributions: if a distribution FX is smaller than a distribution FY relative to this ordering, then FX is closer to a given reference distribution F than FY. Applications of this ordering are given to the problem of nonparametric density estimation.