Score test


Score test

A score test is a statistical test of a simple null hypothesis that a parameter of interest heta isequal to some particular value heta_0. It is the most powerful test when the true value of heta is close to heta_0.

ingle parameter test

The statistic

Let L be the likelihood function which depends on a univariate parameter heta and let x be the data. The score is U( heta) where:U( heta)=frac{partial log L( heta | x)}{partial heta}.

The observed Fisher information is,

:I( heta) = -frac{partial^2 log L( heta | x)}{partial heta^2}.

The statistic to test H_0: heta= heta_0 is:frac{U( heta_0)^2}{I( heta_0)}

which takes a chi^2_1 distribution asymptotically when H_0 is true.

Justification

The case of a likelihood with nuisance parameters

As most powerful test for small deviations

:left(frac{partial log L( heta | x)}{partial heta} ight)_{ heta= heta_0} geq CWhere L is the likelihood function, heta_0 is the value of the parameter of interest under the null hypothesis, and C is a constant set depending onthe size of the test desired (i.e. the probability of rejecting H_0 ifH_0 is true; see Type I error).

The score test is the most powerful test for small deviations from H_0.To see this, consider testing heta= heta_0 versus heta= heta_0+h. By the Neyman-Pearson lemma, the most powerful test has the form

:frac{L( heta_0+h|x)}{L( heta_0|x)} geq K;

Taking the log of both sides yields

:log L( heta_0 + h | x ) - log L( heta_0|x) geq log K.

The score test follows making the substitution

: log L( heta_0+h|x) approx log L( heta_0|x) + h imes left(frac{partial log L( heta | x)}{partial heta} ight)_{ heta= heta_0}

and identifying the C above with log(K).

Relationship with Wald test

Multiple parameters

A more general score test can be derived when there is more than one parameter. Suppose that hat{ heta}_0 is the Maximum Likelihood estimate of heta under the null hypothesis H_0. Then

:U^T(hat{ heta}_0) I^{-1}(hat{ heta}_0) U(hat{ heta}_0) sim chi^2_k

asymptotically under H_0, where k is the number of constraints imposed by the null hypothesis and

:U(hat{ heta}_0) = frac{partial log L(hat{ heta}_0 | x)}{partial heta}

and

:I(hat{ heta}_0) = -frac{partial^2 log L(hat{ heta}_0 | x)}{partial heta partial heta'}.

This can be used to test H_0.

ee also

*Fisher information
*Uniformly most powerful test
*Wald test
*Likelihood Ratio Test
*Score (statistics)


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