# likelihood ratio test for non-nested hypotheses

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

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

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.
• Thank you for your answer! Then if the alternative hypothesis is $\theta\in\Omega$, then how should we define the likelihood ratio function? This is the case where the null hypothesis is nested within the alternative hypothesis. Feb 5 '20 at 6:52
• The alternative hypothesis cannot be $\theta\in\Omega$. The null and alternative hypotheses cannot be nested.