Lecture 11 slides: Large sample tests

1 Likelihood Ratio Test Let X1, ..., Xn be a random sample from a distribution with pdf f(x|θ) where θ is some one dimensional (unknown) parameter. Suppose we want to test the null hypothesis, H0, that θ = θ0 against the alternative hypothesis, Ha, that θ = θ0. Assume the same regularity conditions hold as in the MLE theory. Then likelihood ratio test (LRT) statistic is λ(x) = L(θ0|x) L(θ̂ML|x) where x = (X1, ..., Xn) and θ̂ML is the ML estimator. Then we have Theorem 1. Under the null hypothesis,−2 log λ(x) ⇒ χ1. Proof. Denote ln(θ) = logL(θ|x). By the Taylor theorem, for some θ between θ0 and θ̂ML,