Jensen's inequality for medians

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Abstract

We prove an analogue of Jensen's inequality, with medians instead of means. A novel definition of a median is given, which allows a natural extension to higher dimensions.

Section snippets

An analogue of Jensen's inequality for medians

For any random variable X with a finite expectation EX and for any convex function f, it holds that f(EX)Ef(X). This is one version of famous Jensen's inequality, which plays a significant role in probability and statistics. It is natural to ask if the expectation can be replaced with some other kind of mean value. In this work, we present an analogue of Jensen's inequality, where expectation is replaced by median, and we show that this inequality holds for a class of functions that contains

A characterization of median

Theorem 2.1

For any random variable X, the set of its medians (a point or a closed interval) coincides with the intersection of all closed intervals I that have the following property: If J is any closed interval such that I is a proper subset of J, then P(J)>12.

Proof

Let IM denote the set of all medians of X. It is proved in (ii) of Lemma 1.2 that IMI, for every closed interval I that has a property as stated in the theorem. On the other hand, if IM=[a,b], ab, then intervals (-,b] and [a,+) both have the

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A major part of this research was done during the author's visit to the Department of Statistics at Federal University of Rio de Janeiro (UFRJ), Brasil.

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