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Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-X|^3). Does this statistic have a name, or any nice properties?

Edit: Earlier versions of this question had |m-EX| where |m-X| was; this isn't what I meant.

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I'm still confused. Are you asking about the quantity $E(|m-X|^3)$ or the $m$ that minimizes $E(|m-X|^3)$? – Deane Yang Feb 11 '10 at 3:15
Both, actually. I needed an upper bound on E(|m-X|^3), but the quantity m itself -- which is some sort of measure of central tendency analogous to the mean or the median -- might also be of interest. (I found a way around this in the proof, though, so the question is purely for curiosity's sake now.) – Michael Lugo Feb 11 '10 at 15:38
up vote 5 down vote accepted

The minimizer $m$ is the nearest point projection of $X$ onto the subspace of $L^p$ formed by the constant functions ($p=3$ in your case). This $m$ is sometimes called the $p$-prediction or $p$-predictor of $X$. Apparently, this terminology began with Andô and Amemiya. Some of later papers are Landers and Rogge (who wrote a few other papers, e.g. this one), and Cuesta and Matrán. The term "generalized (conditional) expectation" also appeared.

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I assume you mean |m-X| as opposed to |m-EX|? Otherwise, |m-EX| is not a random variable, so E(|m-EX|^k) = |m-EX|^k is always zero (and hence minimized) when m = EX -- i.e., the mean -- and that's probably not what you're asking.

After a bit of Googling around, it looks like you might be talking about the third absolute central moment E(|X-EX|^3), which is related to something called the Barry-Esseen inequality ... see here.

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Yes, you're right. I meant |m-X|. – Michael Lugo Oct 18 '09 at 23:23

E(|X-EX|^k) is called the k-th central (or centered) moment of the random variable X.

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let f(x) be the probability density function of X. We can define the right and the left and right hand sided moments of X with respect to m as follows:

Left hand one sided k-th moment of x with respect to m = int_[-inf m] (m-x)^k f(x) dx

Right hand one sided k-th moment of x with respect to m = int[m inf] (x-m)^k = f(x) dx

One observes the following analogies

  1. The median is the statistic for which the zeroth left and right hand one sided moments are equal (the zeroth moments are just probabilities)

  2. For the mean, the first left and right hand one sided moments are equal.

  3. For the statistic defined in the question, the second left and right hand one sided moments are equal.

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The mode is also the limiting case m->0; this was pointed out to me at – Terry Tao Oct 26 '09 at 17:10

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