Score Test - Multiple Parameters

Multiple Parameters

A more general score test can be derived when there is more than one parameter. Suppose that is the maximum likelihood estimate of under the null hypothesis . Then

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

asymptotically under, where 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

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

This can be used to test .

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