Score test

Rao's score test, or the score test (often known as the Lagrange multiplier test in econometrics) is a statistical test of a simple null hypothesis that a parameter of interest is equal to some particular value . It is the most powerful test when the true value of is close to . 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.

Single parameter test

The statistic

Let be the likelihood function which depends on a univariate parameter and let be the data. The score is defined as

The Fisher information is

The statistic to test is

which has an asymptotic distribution of , when is true.

Note on notation

Note that some texts use an alternative notation, in which the statistic is tested against a normal distribution. This approach is equivalent and gives identical results.