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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 $\beta$?

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  • $\begingroup$ If you're handed $(x,y)$, what information about $\alpha$ and $\beta$ is there in $y$ that's not already in $x$? As such, why wouldn't you just use the usual MLE for the parameters on the $x$-sample? If you only had $y$, that would be a less trivial question. However, you can estimate the variance and the degrees of freedom from the resulting $t$-distribution of $y$, and hence obtain estimates of $\alpha$ and $\beta$. $\endgroup$ – Glen_b Jul 23 '13 at 23:00
  • $\begingroup$ @Glen_b, you're right. I misstated the problem. The sample $y$ from the Gamma is corrupted by gaussian noise, so that $y \sim \textrm{Gamma}(\alpha,\beta)+N(0,\delta)$ $\endgroup$ – user28389 Jul 24 '13 at 2:21
  • $\begingroup$ Do you mean that $x$ is corrupted? In your original, $y$ is normal, not gamma. It makes even less sense now than before. Could you please edit your question to clearly reflect the actual circumstances? Do you need a third variable? $\endgroup$ – Glen_b Jul 24 '13 at 6:26
  • $\begingroup$ @Glen_b, right again. I meant that $x$ is corrupted. Edited to reflect this change. $\endgroup$ – user28389 Jul 25 '13 at 0:50
  • $\begingroup$ Well, there are definitely ways to estimate the parameters; three examples: (i) you could use ML; (ii) you could take a Bayesian approach; or (iii) you could use Method of Moments (MoM), just for starters. Consider MoM for a second: $\text{E}(Z)\text{Var}(Y)$ should be just a function of $\alpha$, so the MoM estimate of $\alpha$ would be obtained from setting that function of $\hat\alpha$ equal to $\bar z s^2_y$ and solving for $\hat\alpha$. An estimate of $\beta$ can then be backed out from $E(Z)$ or $\text{Var}(Y)$. Finally, an estimate of $\delta$ can then be backed out of $\text{Var}(Z)$. $\endgroup$ – Glen_b Jul 25 '13 at 5:50

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