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I have an equation as follows

\begin{equation} \begin{alignedat}{1} \left(-\frac{\mu_2(\alpha)}{2}\right)^3+ \left(\frac{\mu_3(\alpha)}{2}\right)^2 \end{alignedat} \end{equation} where $\mu_2(\alpha)$ is the second central moment of a set of variables $\alpha$ and $\mu_3(\alpha)$ is the third central moment of the same set of variables. (This equation actually lives under a square root, but that may be irrelevant...)

I'm wondering if this is itself a statistic function?

I've tried relating it to the sixth central moment and kurtosis, but with no luck as yet.

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    @Sasha It seems that your comment indeed answered the question, and should be posted as an answer.2013-06-20

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The expression is related to $\beta_1(\alpha)$ (see this), namely $ \left(-\frac{\mu_2(\alpha)}{2}\right)^3+ \left(\frac{\mu_3(\alpha)}{2}\right)^2 = \frac{1}{8} \mu_2^3(\alpha) \left(\beta_1 - 1 \right)$

Parameter $\beta_1$ is the square of the skewness parameter, i.e. $\beta_1 = \gamma_1^2$. The parameter can not be expressed through higher moments (i.e. 6-th and 4-th as you tried, unless some specific distribution is being considered).