Is there an analytic formula for the kurtosis of a (student t) mixture distribution? The mixture distribution should be composed of noncentral scaled student t components.
The mean of the mixture distribution can be calculated easily just by weighting the component means according to the mixing law. I also could derieve a tractable analytic formula for the mixture variance by means of the "law of total variance" in a straightforward way. 
But for the kurtosis I found no low-level "shortcuts" to get a general analytic formula in this case.
Does anyone have seen such a formula in a book? Any suggestions for a straightforward derivation of such a formula would also be very helping!
Thanks a lot in advance!! Jo
 A: You can apply the law of total expectation to $E(Y^k)$, for $k=4,3,2,1$ and then obtain* a formula for $E[(Y-\mu_Y)^4]$. To get the kurtosis of course you still have to divide by $\sigma_Y^4$
* the conversion from raw to central moments at that point is straightforward, but to save you some labour, this can just be looked up
[If you desire it, you could derive a law of total fourth moments. However, I don't expect it will be especially enlightening, but in fact rather unwieldy.]
So, basically, take the formula you already have for $E(Y)$ in terms of the components, and put $k^\text{th}$ powers everywhere (since the formula should work just as well for the random variable $Y^* = Y^k$ as it does for the random variable $Y$). You now have a formula for the raw moments.
You then use $\mu_4 = \mu'_4 - 4 \mu \mu'_3 + 6 \mu^2 \mu'_2 - 3 \mu^4\,$ to obtain the 4th central moment, which you divide by the square of the variance you already have.
In practice I wouldn't actually derive a direct formula, but simply compute the raw moments, and then compute the central moments as needed. (In some circumstances numerical issues with the potential for cancellation in the above formula may need to be considered.)
