Λ_{n} = \frac{s u p _{θ \in Θ_{0}} L _{n} ( θ )}{s u p _{θ \in Θ_{1}} L _{n} ( θ )}, W_{n} = - 2 lo g Λ_{n} .

Under the null, Wilks theorem gives

W_{n} \Rightarrow χ_{d_{1} - d_{0}}^{2}

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-

α

test rejects for

W_{n}

exceeding the upper

α

quantile of that chi-square law. The approximation is for large samples; boundary parameters, singularities or nonidentifiability can give a different limiting distribution.