Where does that 2 come from in the likelihood ratio test?

This afternoon, in class, we’ve seen Wald test, the likelihood-ratio test, and finally the score test. All of them rely on the same idea and then, use that if with , we can write Or – slightly more interesting – if , then Then one can get that Based on that property, we can derive Wald statistics, that can be visualized below The score test is a test on the square of the slope The idea for the likelihood ratio test is to consider Observe that  can be written, using Taylor’s expansion

for some . The first term is null, since the maximum likelihood estimator is precisely at the maximum of the (log) likelihood. So

That’s more or less where the 2 comes from. Then observe that

and therefore

This test will be discussed further next week (since it is related to Neyman-Pearson’s theorem), but also, that result can be used to derive confidence intervals. With a log-likelihood as follows it is possible to get a confidence interval for the parameter by looking for‘s such that We will discuss that idea later on, in the context of profile likelihood.