# 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 $U\left( heta\right)$ where:$U\left( heta\right)=frac\left\{partial log L\left( heta | x\right)\right\}\left\{partial heta\right\}.$

The observed Fisher information is,

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

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

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

:$left\left(frac\left\{partial log L\left( heta | x\right)\right\}\left\{partial heta\right\} ight\right)_\left\{ heta= heta_0\right\} geq C$Where $L$ 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 $H_0$ if$H_0$ 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

:$frac\left\{L\left( heta_0+h|x\right)\right\}\left\{L\left( heta_0|x\right)\right\} geq K;$

Taking the log of both sides yields

:$log L\left( heta_0 + h | x \right) - log L\left( heta_0|x\right) geq log K.$

The score test follows making the substitution

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

and identifying the $C$ above with $log\left(K\right)$.

Relationship with Wald test

Multiple parameters

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

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

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

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

and

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

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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