Yes Shunday, that's a good insight too. For this case of comparing 2 distributions using the loglikelihood ratio (LLR?) or difference of loglikelihoods, I can see that the distribution used for Ho may be way off of what the fitted parameters would be with this sample, but it may be a different story for another sample.
In this case, let's say that double the difference would be 7 for this new sample, instead of 4.94, let's say that Ho has even more different parameters, I would reject Ho at 1% and can't reject at 0.5%.
So, let's say that the Ho distribution is the "book" or "historic" distribution based on perhaps a larger sample, and I am told to use it, unless (the fitted for theta, using alpha=3) parameter distribution H1 is significant at the 2.5% significant level. If I get the first sample and get 2delta = 4.96, it would meet the requirement and I would use the historic distribution. In the second sample example, I get 2delta = 7 and therefore it would make it rarer for Ho to be true, and I reject Ho for this second sample and therefore the fitted distribution is relevant and I would use it instead.
I hope this reflect the actual use of the test. And explain the case of the axemurderer.
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