Discussion
Consider the problem of testing \(H_{0} : \theta = \theta_{0}\) against \(H_{1} : \theta \ne \theta_{0}\), where \(\theta_{0}\) is specified value of \(\theta\) and is the parameter of one parameter Cramer family. If \(\lambda(x)\) is the likelihood ratio statistic based on sample of size \(n\),
Answer: C
\(-2\log\lambda(x) \to \chi^{2}_{1}\) for \(\theta = \theta_{0}\), as \(n \to \infty\).
This is Wilks' theorem. Under regularity conditions (the Cramer conditions), when the null hypothesis is true the statistic \(-2\log\lambda\) converges in distribution to a chi-square with degrees of freedom equal to the number of parameters fixed by \(H_{0}\), here one.
The result holds only under \(H_{0}\): under a fixed alternative the statistic grows without bound, so 'for all \(\theta\)' in A and B is wrong, and B also has \(n\) degrees of freedom. The limit is a chi-square, not a standard normal (D); its square root, signed, would be normal.
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