Jensen's inequality for medians
Section snippets
An analogue of Jensen's inequality for medians
For any random variable with a finite expectation and for any convex function , it holds that . 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 that have the following property: If J is any closed interval such that I is a proper subset of J, then . Proof Let denote the set of all medians of . It is proved in (ii) of Lemma 1.2 that , for every closed interval that has a property as stated in the theorem. On the other hand, if , , then intervals and both have the
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Cited by (12)
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2022, SIAM Journal on Scientific Computing
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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.