Suppose that $X$ is a random variable with finite first moment and median $m$. Let $X'$ be an independent copy of $X$. What inequalities relate $E|X-X'|$ and $E|X-m|$? What is the best lower bound on $E|X-X'|$ in terms of $E|X-m|$? We easily see that $E|X-X'|\ge E|X-m|$. Can this inequality be improved to $E|X-X'|\ge cE|X-m|$ for some $c>1$? If not, what additional assumptions are need for this to hold?
1 Answer
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Let $X$ be a coin flip, 0 or 1 with equal probability. Then $E[|X-X'|]=1/2$, and $E[|X-m|]=1/2$. The same holds up to within $\epsilon$ for continuous symmetric distributions concentrated near 0 and 1. So we can not improve on the $c$ in the question.
The general subject here is order statistics, where $E[|X-X'|]$ is also known as $2 \lambda_2(X)$, and $\lambda_2$ is the second $L$-moment.