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I extended this question here. Do you mind sharing your thoughts there as well @Glen_b ? And can we agree that out of the 4 tests the KS one is the "most" appropriate for this purpose or do you think I should opt for any of the other 3?
@Glen_b I actually meant exact as in, for instance, the KS test is exact for continuous variables whilst it's conservative for discrete variables. And, exponential random variables are continuous so I thought it's better than the other 3. It's clear that there are many more tests out there I just wanted to simply the list a little into the 4 more "general" or "conventional" ones without being over-elaborate.
I added more context as per you request @Glen_b but, honestly, I have no clue where the 19 came from it's what wolfram alpha does went inverting $Q$ the regularized incomplete gamma function and as far as I know it's correct.
With "test the hypothesis exactly" I meant in the sense of having certain conditions and criteria that are met such as, for example, critical value or p value. It's harder to tell by the eye in contrast numbers don't lie if you know what I mean @Glen_b
Yes, you are right, LRT is very powerful. I don't want anything too manual like QQ-Plot nor anything too complex like LRT. The KS test is one of the most useful, exact and general goodness of fit method which is why I thought it will do the job perfectly fine. Out of the 4 I've listed, you think KS is the way to go @Glen_b ?
@Glen_b it's bayesian estimation not linear regression. It starts with i.i.d. $X_1,...,X_n \sim Exp(\lambda)$ and I had to come up with $X∼Gamma(10,1/12)$ after computing Jeffreys prior and so on.
Instead of $𝑄(10,\alpha/12)$ @Glen_b it should actually be $𝑄(10,12\alpha)$. That value seems more plausible and rather correct to me. And that's what I meant with inverse $\beta$ instead of $12$ it should be $1/12$. But it really depends on your definition it's a matter of scaling.
