For a dataset $X_1, ..., X_n$, the random variables are independent and $X_i \sim \text{EXP}(\theta_i)$. Consider the test where $H_0$ claims $\theta_1 = \theta_2 = ... = \theta_n = \theta$, while $H_A$ claims $H_0$ is false.
Determine the likelihood ratio test and show that it is distribution free under $H_0$.
After some work, if $f(t; \theta_i) = \frac{1}{\theta_i} e^{-t/\theta_i}$ for $t \geq 0$, the likelihood ratio statistic comes out to be $\lambda(x_1, ..., x_n) = \frac{\prod_{i = 1}^n x_i}{\hat \theta^n}$ and the test is to reject $H_0$ if $\frac{\prod_{i = 1}^n x_i}{\hat{\theta}^n} \leq k$ for an appropriately chosen $k$.
Now the question is showing that this test is distribution free under $H_0$. I'm not even sure what is meant by "distribution free under $H_0$". When I search for "distribution free" on the Internet I get references to nonparametric tests. I'd be shocked to discover this test is a non-parametric test (it made an assumption that the data follows exponential distributions), neglecting how I would even show that was true.