Rao's score test, or the score test (called the Lagrange multiplier test in econometrics), is a statistical test of a simple null hypothesis that a parameter of interest θ is equal to some particular value θ 0. It is the most powerful test when the true value of θ is close to θ 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 makes testing feasible when the unconstrained maximum likelihood estimate is a boundary point in the parameter space.
Let L be the likelihood function which depends on θ parameter and let x be the data. The score is:
The observed information is:
Suppose that θ 0 is the maximum likelihood estimate of θ under the null hypothesis Η 0 : θ= θ.
Then
asymptotically under Η 0, where k is the number of constraints imposed by the null hypothesis.