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Suppose you want to estimate the parameters $\alpha$ and $\beta$ of a $\text{gamma}(\alpha, \beta)$ distribution where we know that $\alpha = \beta$. Would you treat this as a distribution with one or two unknown parameters? It seems that you can treat it as a distribution with 1 unknown parameter.

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Are you saying that you know the mean of your gamma distribution is $1$? If so then yes, you can treat the shape and scale parameters as a single unknown, subject to that constraint. –  Henry Apr 7 '12 at 17:36
@Henry: So if you have a $N(a, a^2)$ distribution with $a$ unknown, then if you find an estimator for $a$ you have found an estimator for $a^2$? –  jameie ross Apr 7 '12 at 17:45
yes, and again in your $N(a,a^2)$ example it is subject to a constraint. –  Henry Apr 7 '12 at 17:50
@Henry: I see. So you can find an estimator for the scale parameter (or shape parameter) and say that you have also found an estimator for the shape shape parameter (scale parameter)? –  jameie ross Apr 7 '12 at 18:02
@Henry: In other words, you could say that the shape parameter is known and the scale parameter is unknown? –  jameie ross Apr 7 '12 at 18:09

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