 Solution

For the simple null, the restricted maximum-likelihood estimator is $\tilde{\theta }={\theta }_0$, and henceThe individual scores are independent, have mean zero by part (ii), and have covariance $I_1({\theta }_0)$. The multivariate central limit theorem givesAfter multiplying by $I_1({\theta }_0{)}^{-1/2}$, the limit is standard normal in ${\mathbb{R}}^p$. The continuous mapping theorem therefore yields