Jensenʹs inequality for multivariate medians

Given a probability measure μ on Borel sigma-field of R d , and a function f : R d ↦ R , the main issue of this work is to establish inequalities of the type f ( m ) ⩽ M , where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure μ f = μ ○ f − 1 . For the most popular choice of halfspace depth, we prove that the Jensenʹs inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in [Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461–482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in R d and we present a version of Jensenʹs inequality for such medians. Replacing means in classical Jensenʹs inequality with medians gives rise to applications in the framework of Pitmanʹs estimation.