Article ID Journal Published Year Pages File Type
4619103 Journal of Mathematical Analysis and Applications 2010 12 Pages PDF
Abstract

Given a probability measure μ on Borel sigma-field of Rd, and a function f:Rd↦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 μf=μ○f−1. For the most popular choice of halfspace depth, we prove that the Jensen's 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 Rd and we present a version of Jensen's inequality for such medians. Replacing means in classical Jensen's inequality with medians gives rise to applications in the framework of Pitman's estimation.

Related Topics
Physical Sciences and Engineering Mathematics Analysis