Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$
P.S: The Likelihood-ratio Test consist to consider the critic region $A^*$ as $$A^* = \left[ x \in \Omega : \frac{\sup_{\theta \in \Theta_o} L_{x}(\theta)}{\sup_{\theta \in \Theta} L_x(\theta)} \leq c_\alpha\right] $$
where c is a constant the depends of $ \alpha $ (error type I) and $L$ is the likelihood function.
What is the point of testing $ \theta = 0$ since the exponential distribution is the gamma distribution with parameters $1, \theta$, which is defined for $ \theta > 0$?