TITLE: What m minimizes E(|m-X|^3) for a random variable X?
QUESTION [10 upvotes]: 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.

REPLY [5 votes]: 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.