Suppose under $H_0$, a test statistic $T$ has a gamma distribution with paramaters $\theta$ and $k$. Suppose also that the distribution of $T$ under $H_1$ is unknown. What is the appropriate rejection region in this case? Is it: $$ T \geq t_\alpha$$ or $$\hat{T}\leq t_{1-(\alpha/2)} \hspace{0.3cm}\cup \hspace{0.3cm} \hat{T}\geq t_{\alpha/2} $$ where $\hat{T}$ is the value of the statistic $T$ for a sample $X_1, X_2,\ldots, X_n$ and $t_\alpha$ is the critical value of the Gamma distribution with parameters $\theta$ and $k$.
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2$\begingroup$ Would depend on what $H_0$ and $H_1$ are. $\endgroup$– StubbornAtomCommented Mar 13, 2019 at 11:14
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$\begingroup$ @StubbornAtom, Would you please elaborate? $\endgroup$– NooobCommented Mar 13, 2019 at 11:42
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1$\begingroup$ If you are testing the parameters, the critical region would depend on whether this is a left/right/both-tailed test. $\endgroup$– StubbornAtomCommented Mar 13, 2019 at 11:52
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$\begingroup$ @StubbornAtom, Let's say for example, the null hypothesis $H_0$ is that the sample $X_1,X_2, \ldots, X_n$ is a white noise, and $H_1$ is that the sample is not a white noise, The statistic $T$ has a $Gamma(\theta,k)$ under $H_0$, but under $H_1$ the distribution is unkown. How would you choose the critical region in this case? $\endgroup$– NooobCommented Mar 13, 2019 at 12:09
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$\begingroup$ Since the distribution of $T$ under $H_1$ is unknown, the rejection region better include the entire negative real line: the observation of a negative value of $T$ is a dead giveaway that $H_0 $ cannot possibly be true and so must be rejected. $\endgroup$– Dilip SarwateCommented Mar 13, 2019 at 12:56
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