Hypothesis test for composite null hypothesis of exponential parameter I'm having trouble defining the reject region based on the generalized likelihood ratio test. This is from a question of past exam I'm self-studying and still have the doubt, given that I got it wrong.
Let $X_{1},\ldots,X_{n}$ a sample from an exponential distribution $X \sim \text{Exp}(\theta)$ with $\theta = \mathbb{E}[X]$. The parameter space is $\Omega = \{\theta \mid\theta>0\}.$ 
I need to contrast the next hypothesis: $H_{0}:\theta \geq \theta_{0}$ vs $H_{1}: \theta< \theta_{0}$.
The set of the null hypothesis is: $\Omega_{0} = \{\theta \mid\theta \geq \theta_{0}\}$ and the set of the alternative hypothesis is: $\Omega_{1} = \{\theta \mid\theta < \theta_{0}\}$. So that $\Omega = \Omega_{0} \cup\Omega_{1}.$
The likelihood function for $\theta$ is given by: $$ L(\theta) = \theta^{-n} \exp\{ -\theta^{-1} \bar{X} \} $$
I know the generalized likelihood ratio is given by $$\Lambda = \frac{\sup_{\theta \in \Omega_{0}} L (\theta)}{\sup_{\theta \in \Omega} L (\theta)}$$
and serves as a test statistic. For the denominator, we have that the MLE of $\theta$ is given by $\hat{\theta} = \bar{X}$, and with this the denominator is $$L(\hat{\theta}) = \bar{X}^{-n}\exp\{ -\bar{X}^{-1}n\bar{X} \} = \bar{X}^{-n}\exp\{ -n \}$$
My problem is: how do I solve for the numerator $\sup_{\theta \in \Omega_{0}} L (\theta)$ ? Because I already know how to solve the test given a simple null hypothesis $H_{0}: \theta = \theta_{0}$ and I'm really stucked with this problem.
Thank you for your help.
 A: Unrestricted MLE of $\theta$ is as you say $\hat\theta=\overline X$, the sample mean.
Now under the restriction $\theta\ge\theta_0$, argue that MLE of $\theta$ must be $$\hat{\hat\theta}=\begin{cases}\hat\theta&,\text{ if }\hat\theta\ge\theta_0 \\ \theta_0&,\text{ if }\hat\theta<\theta_0\end{cases}$$
So depending upon whether $\overline X\ge \theta_0$ or $\overline X<\theta_0$, the likelihood ratio statistic takes the form
\begin{align}
\Lambda=\frac{\sup_{\theta\ge\theta_0} L(\theta)}{\sup_{\theta}L(\theta)}&=\frac{L(\hat{\hat\theta})}{L(\hat\theta)}
\\&=\begin{cases}1&,\text{ if }\hat\theta\ge\theta_0 \\\\ \frac{L(\theta_0)}{L(\hat\theta)}&,\text{ if }\hat\theta<\theta_0\end{cases}
\end{align}
Now it is a matter of studying this ratio as a function of $\overline X$ when $\hat\theta<\theta_0$. Remember to reject $H_0$ for small values of $\Lambda$. The case corresponding to $\hat\theta\ge\theta_0$ leads to trivial acceptance of $H_0$. 
A: Test. Once you figure out that you want to reject for small $\bar X,$ the task remains to find the critical value $c$ for a test of $H_0: \theta \ge \theta_0$ vs $H_a: \theta < \theta_0$ at level $\alpha = 0.05.$ That is, $P(\bar X \ge c\,|\,\theta_0) = 0.05.$
Null distribution Suppose that $\bar X$ is the mean $\bar X$ of a random sample $X_1, X_2, \dots, X_n$ from $\mathsf{Exp}(\theta_0),$ so that $E(X_i) = \theta.$
Then you can use moment generating functions to show $\bar X \sim \mathsf{Gamma}(\text{shape} = n, \text{scale}=\theta/n).$
Thus $E(\bar X) = \theta$ and $SD(\bar X) = \theta/\sqrt{n}).$
The following simulation in R of a million samples of size $n =25$ from $\mathsf{Exp}(\text{mean} = \theta = 50)$ $\equiv$ $\mathsf{Exp}(\text{rate} = \lambda = 0.02)$
compares a histogram of the simulated distribution of means $\bar X$ with the density function of $\mathsf{Gamma}(n, \text{scale} = \theta/n)
\equiv \mathsf{Gamma}(n, \text{rate} = n/\theta).$ (Perhaps see Wikipedia on gamma distributions; R uses the rate parameter.)
set.seed(2019)
n = 25;  th = 50;  lam = 1/th
a = replicate( 10^6, mean(rexp(n, lam)) )
lbl = "Simulated Sums of Exponential Data with Gamma Density"
hist(a, prob=T, br = 50, col="skyblue2", main = lbl)
  curve(dgamma(x, n, n/th), 0, 120, add=T, lwd=2)


Critical value. Then, testing $H_0: \theta \ge \theta_0 = 50$ vs. $H_a: \theta < 50,$ based on $n = 25$ exponential observations, we can use R to find the critical value as $c = 34.764.$ That is, 
$P(\bar X \le 34.764\, |\, \theta_0 = 50) = 0.05.$ 
th.0 = 50;  n = 25
c = qgamma(.05, n, n/th.0);  c
[1] 34.76425

Power. If we happen to sample from an exponential distribution with mean $\theta_a = 40$ (rate $\lambda_a = 0.025),$ what is the probability of rejecting $H_0?$ The result from R is
only $P(\bar X \le 34.764\, |\, \theta_a = 40) = 0.268.$ So using this test, $n = 25$ observations is not enough reliably to discover if the true population mean is $\theta_a = 40$ instead of $\theta_0 = 50.$
However, the power against the alternative value $\theta_a = 30$ is
almost 80%.
n = 25; th = 40; c = 34.764
pwr = pgamma(c, n, n/th);  pwr
[1] 0.2682961

n = 25; th = 30; c = 34.764
pwr = pgamma(c, n, n/th);  pwr
[1] 0.7942828

