Score test

The score test is a statistical test of a simple null hypothesis (that the parameter of interest $$\theta$$ is equal to some particular value $$\theta_0$$):

\left(\frac{\partial \log L(\theta | x)}{\partial \theta}\right)_{\theta=\theta_0} \geq C $$ Where $$L$$ is the likelihood function, $$\theta_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 $$\theta=\theta_0$$ versus $$\theta=\theta_0+h$$. 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 $$\log(K)$$.

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 $$\theta$$ under the null hypothesis $$H_0$$. Then



U'(\hat{\theta}_0) I^{-1}(\hat{\theta}_0) U(\hat{\theta}_0) \sim \chi^2_k $$

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



U(\hat{\theta}_0) = \frac{\partial \log L(\hat{\theta}_0 | x)}{\partial \theta} $$

and



I(\hat{\theta}_0) = -\frac{\partial^2 \log L(\hat{\theta}_0 | x)}{\partial \theta \partial \theta'}. $$

This can be used to test $$H_0$$.