I want to perform a statistical test given two simple hypothesis. I have the analytical form of my pdf in dependence of $3$ parameters which I know are fixed at $\alpha = 0.65, \beta = 0.06, \gamma = -0.18$ for one hypothesis and $\alpha = 1/3, \beta = 0, \gamma = 0$ for the other hypothesis. For the second hypothesis the pdf reduces to a uniform pdf.
I have a dataset of $50.000$ points sampled from the first distribution (the not uniform one) so the test is trivial but still, I have to do it.
I compute the value of each pdf at every point of the dataset, I take their logarithm and then I subtract them and sum the result to obtain $\log(\lambda)$. Now I find a very low result meaning $\exp(-40772)$ which clearly support the nonuniform distribution of data but how can I compute the significance region or the power of the test? I cannot use Wilk's theorem since my pdf is completely defined. The only thing I can do is the following:
$P(\lambda < c | H_0) = \alpha$
But I do not know how to compute $c$. Has anyone suggestions?
For clarity I will post the two pdfs:
Variables $\theta, \phi$ defined in $\theta \in [0, \pi], \phi \in [0, 2\pi]$.
$$ (,)=34[0.5(1−)+(0.5)(3−1)()^{2}−()^2(2)−\sqrt{2}\gamma(2)()] $$
$$\alpha = 0.65, \beta = 0.06, \gamma = -0.18 $$
Uniform pdf: $$ U(\theta, \phi) = \begin{cases} \frac{1}{2\pi^{2}} & \theta \in [0, \pi], \phi \in [0, 2\pi]\\ 0 & otherwise \end{cases} $$