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I'm trying to work out how to apply the method of moments to estimate the parameters of the skew-t distribution.

From slide 12 of the following presentation: http://www.eief.it/files/2008/11/monti-8-maggio.pdf or page 17 of Azzalini and Capitanio 2012 available here: http://azzalini.stat.unipd.it/SN/se-ext.pdf the first four moments are:

$$ \mu = \xi + \omega \delta b_v \\ \sigma^2 = \omega^2\left\{\frac{\nu}{\nu - 2} - \delta^2b_v^2\right\}\\ \gamma_1 = \delta b_v \left\{ \frac{\nu(3-\delta^2)}{\nu-3}-\frac{3\nu}{\nu-2} + 2\delta^2b_v^2\right\}\left\{\frac{\nu}{\nu-2}-\delta^2b_v^2\right\}^{-3/2} \\ \gamma_2 = \left\{\frac{3\nu^2}{(\nu-2)(\nu-4)}-\frac{4\delta^2b_v^2\nu(3-\delta^2)}{\nu-3}+\frac{6\delta^2b_v^2\nu}{\nu-2}-3\delta^4b_v^4\right\}\left\{\frac{\nu}{\nu-2}-\delta^2b_v^2\right\}^{-2}-3 $$

where

$$ b_v = \left(\frac{\nu}{\pi}\right)^{1/2}\frac{\Gamma(\nu/2+1/2)}{\Gamma(\nu/2)} \\ \delta = \frac{\alpha}{(1+\alpha^2)^{1/2}} $$

and $\xi$, $\omega$, $\alpha$ are the location, scale and skewness parameters and $\nu$ the degrees of freedom.

For the skew-normal distribution i.e. the special case of $\nu=\infty$, it's easy to proceed as described by the Wikipedia article on that distribution (https://en.wikipedia.org/wiki/Skew_normal_distribution#Estimation) and relate the sample skewness to the skewness parameter; once the skewness parameter is estimated then the others follow.

However, I'm finding it challenging to rearrange the above equations to derive the skewness parameter (via $\delta$) either using the skewness or kurtosis. My intuition is that this is not possible due to the skewness depending not just on the skewness parameter but also the degrees of freedom. Is this correct?

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You have four (highly) non-linear equations and four unknowns. But also, note that the last two equations do not depend on the location parameter $\xi$, or on the scale parameter $\omega$. So these last two form an autonomous sub-system. To minimize the dimensions, first you calculate from the sample $\hat \gamma_1,\;\hat \gamma_2$ and then you ask from a software to solve the two-equations system for you, where you code in fundamentally $\alpha$ and $\nu$, using the $\delta, b_v$ as convenient functions of the parameters of interest.

In other words, $\alpha$ and $\nu$ are estimated/calculated jointly. I cannot say whether a unique solution is guaranteed.

Deriving a theoretical, non-implicit equation of the form

$$\alpha = h(...)$$

with the $\alpha$ absent from the right hand side, does not appear feasible. If you have access to software that solves symbolic equation systems (i.e without any specific numerical values supplied), you could give it a try, to see whether a theoretical expression of that form emerges.

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