Score test

The score test evaluates the likelihood gradient at the restricted estimator and scales its quadratic form by inverse Fisher information.

A restricted maximum-likelihood estimator maximizes likelihood over the null parameter space rather than the full model.

Under a regular simple $p$-parameter null, the normalized score converges to $N_p(0,I_1)$, so its information-standardized squared norm converges to ${\chi }_p^2$.

If $Z\sim N_p(0,I_p)$, then $Z^{T}Z={\sum }_{j=1}^p{Z}_j^2\sim {\chi }_p^2$.

For independent $N(\mu ,{\sigma }^2)$ observations, ${\sum }_i(X_i-\bar{X}{)}^2/{\sigma }^2$ has the ${\chi }_{n-1}^2$ distribution.

If $Q_m\sim {\chi }_m^2$, then $(Q_m-m)/\sqrt{2m}$ converges in distribution to $N(0,1)$.