The Distribution of Likelihood Ratio test statistics
Wilk's Theorem states that when the sample size is large, the distribution of \Lambda under H_{0} approaches a \chi ^2 distribution:
\displaystyle \displaystyle \Lambda \overset {n\to \infty }{\longrightarrow } \chi _ d^2 \qquad \text {where } d = \text {dim}(\Theta ) - \text {dim}(\Theta _0)
where d is the degree of freedom of the \chi ^2 distribution.
Power of Likelihood Ratio Test
The Neyman–Pearson lemma states that among all tests that test for the simple hypotheses H_0: \theta =\theta _0\, ;\, H_ A:\theta =\theta _ A at significance level \alpha, the likelihood ratio test is the most powerful. That is, among all tests testing the same simple hypotheses and at the same significance level, the likelihood ratio test gives the largest probability of rejecting the null when indeed the alternate is true.
Wilk's Theorem
1/2 points (graded)
Given the hypotheses
H_0: \theta = \theta _0; H_ A: \theta \neq \theta _0,
where \theta is a scalar (it takes values from some subset of the Reals).
What is the approximate distribution of the LR-test statistic \Lambda (x) when the sample size is large?
t-distribution
\chi ^2-distribution
Normal distribution
correct
What is the degrees of freedom d of the distribution? (If the degrees of freedom doesn't apply, enter -1.)
d=\quad