Likelihood ratio test for a random variable following the Gamma Distribution

Assuming we have a random variable $X\sim \operatorname{Gamma}\left(\alpha,\beta \right )$:$\frac{1}{\Gamma (\alpha )\beta ^{\alpha}}x^{\alpha-1}e^{\frac{-x}{\beta }}$

I'd like to test the hypothesis: $H_{0}:\beta=\beta_0$ vs: $H_{A}:\beta\neq \beta_0$ using a likelihood ratio test. I am assuming alpha is known and have derived my MLE: $\hat{\beta}_{MLE}=\frac{\bar{x}}{\alpha}$.

I have worked through it and so far and to my understanding the MLE is used as the alternative. So far I have come up with: $D=2\ln \left( \frac{L(\hat{\beta})}{L{\beta_0}}\right )=2\left[n\alpha\ln\left(\frac{\beta_0}{\hat{\beta}} \right )+\sum_{i=1}^{n} x_i\left(\frac{1}{\beta_0}-\frac{1}{\hat{\beta}} \right ) \right ]$

I really have no idea whether or not I have done this correctly. If the above is correct, what is the asymptotic distribution that this statistic converges to as $n \to \infty$?

• Your question asks for the asymptotic distribution of a sum of iid variables, as multiplied by a constant ($2\left(1/\beta_0-1/\hat{\beta}\right)$) to which a constant has been added ($2 n\alpha \log(\beta_0/\hat{\beta})$). Surely you know all about that kind of asymptotic distribution! – whuber May 20 '13 at 14:35