I was reading over the documentation for the Rao's Score/Lagrange Multiplier Test on wikipedia and ran across this:
Rao's score test is a statistical test of a simple null hypothesis that a parameter of interest $\theta$ is equal to some particular value $\theta_0$. It is the most powerful test when the true value of $\theta$ is close to $\theta_0$. The main advantage of the Score-test is that it does not require an estimate of the information under the alternative hypothesis or unconstrained maximum likelihood. This constitutes a potential advantage in comparison to other tests, such as the Wald test and the generalized likelihood ratio test (GLRT). This makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.
In statistics class we were told the Likelihood Ratio test is the best test to evaluate nested logistic regression models as it utilizes more of your data than Wald or the Score test. They mentioned that the other tests could be useful in certain situations, which I guess is being described here.
I've looked through gung's very helpful discussion of the trinity of tests. As far as I can make out the Score method is most useful when the likelihood that the other two tests rely on approaches large numbers/infinity, but am still confused as to what is being said in the above quote. The italicized portion of the quote is what is most confusing.
Could someone break down the language a bit more, talk about thetas, the likelihood and parameter space, and outline in which type of applied situations Rao's score is more powerful? Thanks!