Divergence Measures and Weak Majorization in Estimation Problems

Most of the statistical estimation procedures are based on a quite simple principle: find the distribution that, within a certain class, is as similar as possible to the empirical distribution, obtained from the sample observations. This leads to the minimization of some statistical functionals, usually interpreted ad measures of distance or divergence between distributions. In this paper we study the majorization pre-order of the distance between distributions. This concept, known in literature as relative majorization, is extended to the weak definition of majorization, which is more relevant in many practical contexts such as estimation problem. Providing mathematical proofs, we study under which conditions statistical functionals are consistent with respect to the relative weak majorization (from above) pre-order.