I want to perform a statistical test based on two simple assumptions. I have the analytical form of my pdf based on 3 parameters that I know are set to alpha = 0.65, beta = 0.06, gamma = -0.18 for a hypothesis and alpha = 1/3, beta = 0, gamma = 0 for the other hypothesis. For the second hypothesis, the pdf is reduced to a uniform pdf.

I have a data set of 50,000 points sampled from the first distribution (the nonuniform), so the test is trivial but I still have to do it.

I calculate the value of each pdf at each point of the dataset, take their logarithm, then subtract them and add the result to get the log (lambda). Now, I find a very weak result, which means exp (-40772), which clearly supports the non-uniform distribution of data, but how can I calculate the region of significance or the power of the test? I can not use Wilk's theorem because my pdf is completely defined. The only thing I can do is:

P (lambda <c | H0) = alpha

But I do not know how to calculate c. Anyone have any suggestions?

For clarity, I will post both PDF files:

variables $ theta, phi $ defined in $ theta in [0, pi] phi in [0, 2pi]$.

$$ (θ, φ) = 34π[0.5(1−𝛼)+(0.5)(3𝛼−1)𝑐𝑜𝑠(𝜃)2−𝛽𝑠𝑖𝑛(𝜃)2𝑐𝑜𝑠(2𝜙)−2√𝛾𝑠𝑖𝑛(2𝜃)𝑐𝑜𝑠(𝜙)] $$

$$ alpha = 0.65, beta = 0.06, gamma = -0.18 $$

Uniform pdf:

$$ U ( theta, phi) = begin {case}

frac {1} {2 pi ^ {2}} & theta in [0, pi] phi in [0, 2pi]\

0 & otherwise

end {cases} $$