What m minimizes E(|m-X|^3) for a random variable X? 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.
 A: 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.
A: 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.
A: E(|X-EX|^k) is called the k-th central (or centered) moment of the random variable X.
A: 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


*

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

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

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