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.



Cite this blog post
Arthur Charpentier (2015, October 15). Where does that 2 come from in the likelihood ratio test? Freakonometrics. Retrieved May 23, 2024, from https://doi.org/10.58079/ov0u

4 thoughts on “Where does that 2 come from in the likelihood ratio test?”

  1. Hello, we can’t see well the pictures, maybe because this is an old thread..? Thank you.

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