Null distribution (log likelihood)

determinining a rejection region for a test requires finding a critical value such that $$P(log(LR) > d) = \alpha$$

the critical value is the $$(1-\alpha)$$ quantile of the distribution of log(LR) when the null hypothesis is true.

This is the Null distribution of the log-likelihood ratio test statistics

"So far in this unit, an approach to hypothesis testing has been described based on rejection regions. This approach, which was developed by Jerzy Neyman and Egon Pearson, involves specifying a test size alpha and a rejection region prior to doing the test. An alternative approach, [In M248 this approach is called significance testing.] advocated by Ronald Fisher (and very commonly used), is to evaluate the strength of evidence against the null hypothesis by means of a significance probability, or bold p-value. This is the probability of obtaining data at least as extreme as what has actually been observed, if the null hypothesis were true. The two approaches are really variations on a common theme – though you’d never think so from the heated exchanges between their early protagonists."