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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 ideaand then, use that if   with , we can writeOr - slightly more interesting - if  , thenThen one can get thatBased on that property, we can derive Wald statistics,that can be visualized belowThe score test is a test on the square of the slopeThe idea for the likelihood ratio test is to considerObserve that  can be written, using Taylor's expansionfor some . The first term is null, since the maximum likelihood estimator is precisely at the maximum of the (log) likelihood. SoThat's more or less where the 2 comes from. Then observe thatand thereforeThis 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 followsit is possible to get a confidence interval for the parameter by looking for's such thatWe will discuss that idea later on, in the context of profile likelihood.