likelihood ratio test for non-nested hypotheses

Question

I'm reading the statistics textbook written by Hogg, Tanis & Zimmermann and wanted to ask here if this book is correct about the likelihood ratio test.

In the section about the likelihood ratio test, the book explains how to construct a likelihood ratio function for a test. When we measure a random variable $\bf{X}$ that follows a pdf $f(x|\theta)$ with a parameter $\theta$, and the two competing hypotheses are

\begin{equation} H_{0} : \theta \in \omega \qquad H_{1} : \theta \in \omega^{'} \end{equation} where $\omega^{'}$ is a complement of $\omega$, then according to the book, the likelihood ratio test is done using this likelihood ratio. \begin{equation} \lambda = \frac{L(\hat{\omega})}{L(\hat{\Omega})} \end{equation} where $\Omega$ is the union of $\omega$ and $\omega^{'}$.

But it seems the ration assumes that the null hypothesis is nested within the alternative hypothesis and thus is not appropriate for testing $H_{0}$ against $H_{1}$, because they are non-tested hypotheses. Am I missing something?

Answer 1

No, it is not assumed that null hypothesis is nested within alternative hypothesis.

Here $L(\hat\Omega)=\max(L(\hat\omega),L(\hat\omega'))$ where $L(\hat\omega')$ is the maximum of the likelihood function with respect to $\theta$ when $\theta\in\omega'$. Finding critical region, we select some $k<1$ so that $$\lambda=\frac{L(\hat \omega)}{\max(L(\hat\omega),L(\hat\omega'))}\leq k$$ This is the same as $$ L(\hat \omega) \leq k\cdot \max(L(\hat\omega),L(\hat\omega')) $$ In the case when $L(\hat\omega)>L(\hat\omega')$ this inequality is not fulfilled. So this inequality turnes into $$ L(\hat \omega) \leq k\cdot L(\hat\omega') \quad \text{or}\quad \frac{L(\hat \omega)}{L(\hat \omega')} \leq k. $$ You can see that for $k<1$ two inequalities $$ \lambda=\frac{L(\hat \omega)}{\max(L(\hat\omega),L(\hat\omega'))}\leq k \quad \text{and}\quad \frac{L(\hat \omega)}{L(\hat \omega')} \leq k $$ are equivalent and give the same critical region.