Suppose that we are interested in testing the simple null hypothesis H0 : = 0 versus the

Chapter 10, Problem 10.111

(choose chapter or problem)

Suppose that we are interested in testing the simple null hypothesis \(H_{0}: \theta=\theta_{0}\) versus the simple alternative hypothesis \(H_{a}: \theta=\theta_{a}\). According to the Neyman–Pearson lemma, the test that maximizes the power at \(\theta_{a}\) has a rejection region determined by

\(\frac{L\left(\theta_{0}\right)}{L\left(\theta_{a}\right)}<k\).

In the context of a likelihood ratio test, if we are interested in the simple \(H_0\) and \(H_a\), as stated, then \(\Omega_{0}=\left\{\theta_{0}\right\}, \Omega_{a}=\left\{\theta_{a}\right\}\), and \(\Omega=\left\{\theta_{0}, \theta_{a}\right\}\).

a Show that the likelihood ratio \(\lambda\) is given by

\(\lambda=\frac{L\left(\theta_{0}\right)}{\max \left\{L\left(\theta_{0}\right), L\left(\theta_{a}\right)\right\}}=\frac{1}{\max \left\{1, \frac{L\left(\theta_{a}\right)}{L\left(\theta_{0}\right)}\right\}}\).

b Argue that \(\lambda<k\) if and only if, for some constant \(k^{\prime}\),

\(\frac{L\left(\theta_{0}\right)}{L\left(\theta_{a}\right)}<k^{\prime}\).

c What do the results in parts (a) and (b) imply about likelihood ratio tests when both the null and alternative hypotheses are simple?