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.