Suppose that I have an infinite scale mixture of zero-mean normal distributions, whose mixing distribution is gamma with parameters $\alpha$ and $\beta$. The data is thus distributed according to a generalized Student's $t$ distribution. I am handed a pair of samples, $(z,y)$, generated through the following procedure, and asked to estimate $\alpha$ and $\beta$. First, $x$ is sampled directly from the Gamma distribution. Then it is corrupted by gaussian noise, so that $z = x+N(0,\delta)$. Finally, $y \sim N(0,1/x).$
Are there known estimators for $\alpha$ and $\beta$?