= Solution
The likelihood-ratio statistic for nested parameter spaces is
$$
\Lambda_n=\frac{\sup_{\theta\in\Theta_0}L_n(\theta)}{\sup_{\theta\in\Theta_1}L_n(\theta)},\qquad W_n=-2\log\Lambda_n.
$$
Under the null, <Wilks theorem> gives $W_n\Rightarrow\chi^2_{d_1-d_0}$ when the full model and null submodel are smooth and identifiable, the true parameter is an interior regular point, <Fisher information> is nonsingular in the identifiable directions, and the usual differentiability and likelihood-asymptotic conditions hold. A level-$\alpha$ test rejects for $W_n$ exceeding the upper $\alpha$ quantile of that chi-square law. The approximation is for large samples; boundary parameters, singularities or nonidentifiability can give a different limiting distribution.
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