Score test

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A score test 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 $θ 0$ .

1 Single parameter test
2 Multiple parameters
3 See also

[edit] Single parameter test

[edit] The statistic

Let $L$ be the likelihood function which depends on a univariate parameter $θ$ and let $x$ be the data. The score is $U (θ)$ where

$U(\theta)=\frac{\partial \log L(\theta | x)}{\partial \theta}.$

The Fisher information is,

$I(\theta) = -\frac{\partial^2 \log L(\theta | x)}{\partial \theta^2}.$

The statistic to test $H 0 :θ = θ 0$ is

$\frac{U(\theta_0)^2}{I(\theta_0)}$

which takes a $\chi^2_1$ distribution asymptotically when $H 0$ is true.

[edit] Justification

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[edit] The case of a likelihood with nuisance parameters

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[edit] As most powerful test for small deviations

$\left(\frac{\partial \log L(\theta | x)}{\partial \theta}\right)_{\theta=\theta_0} \geq C$

Where $L$ is the likelihood function, $θ 0$ is the value of the parameter of interest under the null hypothesis, and $C$ is a constant set depending on the 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 $θ = θ 0$ versus $θ = θ 0 + h$ . By the Neyman-Pearson lemma, the most powerful test has the form

$\frac{L(\theta_0+h|x)}{L(\theta_0|x)} \geq K;$

Taking the log of both sides yields

$\log L(\theta_0 + h | x ) - \log L(\theta_0|x) \geq \log K.$

The score test follows making the substitution

$\log L(\theta_0+h|x) \approx \log L(\theta_0|x) + h\times \left(\frac{\partial \log L(\theta | x)}{\partial \theta}\right)_{\theta=\theta_0}$

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

[edit] Relationship with Wald test

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[edit] Multiple parameters

A more general score test can be derived when there is more than one parameter. Suppose that $\hat{\theta}_0$ is the Maximum Likelihood estimate of $θ$ under the null hypothesis $H 0$ . Then