# Definition of the likelihood ratio statistic

The likelihood ratio statistic for testing $$H_0:\theta\in\Theta_0$$ versus $$H_1:\theta\in\Theta_0^c$$ is usually defined by $$\lambda(\mathbf x) =\frac{\sup_{\theta\in\Theta_0}L(\theta\mid\mathbf x)}{\sup_{\theta\in\Theta} L(\theta\mid\mathbf x)},$$ where $$\Theta$$ is the whole parameter space and $$\Theta_0\subset\Theta$$ (see, for example, Secion 8.2.1 of Casella and Berger (2002)).

Why is the supremum in the denominator taken over all $$\Theta$$ instead of $$\Theta_0^c$$? Would it make sense to define the likelihood ratio statistic when the supremum in the denominator is taken over $$\Theta_0^c$$? If the supremum in the denominator was taken over $$\Theta_0^c$$, $$\lambda(\mathbf x)$$ would not necessarily satisfy $$0\le\lambda(\textbf x)\le1$$, but it seems that this choice would be more intuitive since we would compare the likelihood when $$H_0$$ is true with the likelihood when $$H_1$$ is true. Or are these two options ($$\Theta$$ and $$\Theta_0^c$$) actually equivalent?

Any help is much appreciated!

• Using Wilk's theorem, the LR will have an asymptotic $\chi^2$ distribution with the number of degrees of freedom equal to the difference in dimensionality between $\Theta_0$ and $\Theta$. The distribution of the statistic can be used accurately estimate what level of reduction is significant. Apr 7 '20 at 21:19

Suppose $$\boldsymbol X=(X_1,\ldots,X_n)$$ is a random vector whose distribution is parameterized by $$\theta$$, where $$\theta\in \Theta\subseteq \mathbb R^p$$. Let $$L(\theta\mid \boldsymbol x)$$ be the likelihood function given the sample $$\boldsymbol x=(x_1,\ldots,x_n)$$.

In general we consider the problem of testing the null $$H_0:\theta\in \Theta_0$$ against the alternative $$H_1:\theta\in \Theta_1$$, where $$\Theta_0\subset \Theta$$ and $$\Theta_1\subseteq \Theta-\Theta_0$$.

We prefer $$H_0$$ to $$H_1$$ ($$H_1$$ to $$H_0$$) if $$\sup_{\theta}\{L(\theta\mid \boldsymbol x):\theta\in\Theta_0\}>(<) \sup_{\theta}\{L(\theta\mid \boldsymbol x):\theta\in\Theta_1\}$$

When $$H_0$$ is true (false), the ratio $$r(\boldsymbol x)=\frac{\sup_{\theta\in\Theta_0}L(\theta\mid \boldsymbol x)}{\sup_{\theta\in\Theta_1}L(\theta\mid \boldsymbol x)}$$

is expected to be large (small). But $$r$$ is not bounded above.

So we modify $$r(\boldsymbol x)$$ by

$$\Lambda(\boldsymbol x)=\frac{\sup_{\theta\in\Theta_0}L(\theta\mid \boldsymbol x)}{\sup_{\theta\in \Theta_0 \cup \Theta_1}L(\theta\mid \boldsymbol x)}=\frac{\sup_{\theta\in\Theta_0}L(\theta\mid \boldsymbol x)}{\sup_{\theta\in \Theta}L(\theta\mid \boldsymbol x)}$$

If $$H_0$$ is true (false), then as before, $$\Lambda$$ is expected to be large (small).

However we now have $$\Lambda\in (0,1]$$, where we trivially accept (rather fail to reject) $$H_0$$ whenever $$\Lambda=1$$.

This justifies a left-tailed test based on $$\Lambda$$, and $$\Lambda(\boldsymbol X)$$ is called the likelihood ratio criterion.