- Score test
A score test is a
statistical testof a simple null hypothesisthat a parameter of interest isequal to some particular value . It is the most powerful test when the true value of is close to .
ingle parameter test
Let be the
likelihood functionwhich depends on a univariate parameter and let be the data. The score is where:
The observed Fisher information is,
The statistic to test is:
which takes a distribution asymptotically when is true.
The case of a likelihood with nuisance parameters
As most powerful test for small deviations
:Where is the
likelihood function, is the value of the parameter of interest under the null hypothesis, and is a constant set depending onthe size of the test desired (i.e. the probability of rejecting if is true; see Type I error).
The score test is the most powerful test for small deviations from .To see this, consider testing versus . By the
Neyman-Pearson lemma, the most powerful test has the form
Taking the log of both sides yields
The score test follows making the substitution
and identifying the above with .
Relationship with Wald test
A more general score test can be derived when there is more than one parameter. Suppose that is the
Maximum Likelihoodestimate of under the null hypothesis . Then
asymptotically under , where is the number of constraints imposed by the null hypothesis and
This can be used to test .
Uniformly most powerful test
*Likelihood Ratio Test
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