Score test

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The score test is a statistical test of a simple null hypothesis (that the parameter of interest $θ$ is equal to some particular value $θ 0$ ):

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

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] 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