I just learned the likelihood ratio test (LRT) method for model selection and worked out some examples. However, I am still a bit confused with the meaning of it.
Basically, for a family of model parametrised by $\theta$, we consider the restricted family of models parametrised by $\theta_0$ and then we propose the version of restricted parameter space as null hypothesis against the alternative hypothesis which is the general $\theta$ version of models. One then consider the ratio \begin{equation} \text{LR} = -2\log \left[ \frac{\max_{\theta\in\Theta_0} L(\theta)}{\max_{\theta\in\Theta} L(\theta)} \right]. \end{equation} It turns out that this LTR test statistic asymptotically approach the $\chi^2$ distribution and hence one can do hypothesis test to see if one can reject the null hypothesis.
Question: I notice that LRT is useful only in the case of nested parameter spaces, that is, $\theta_0$ must be a restricted version of the general $\theta$. Then, I don't see the point to use LRT for model selection.
If $\theta_0$ is a restricted version of $\theta$, then basically they are "the same model", i.e. parametrised by the same parameter(s) $\theta$. If so, why don't we simply use maximum likelihood estimation to estimate the best parameter $\hat{\theta}$? as usually $\theta_0$ is just some fixed value of $\theta$, then "trying to show $\theta_0$ is not a good choice" seems to not really useful, isn't it? Could someone point out what did I miss? thanks!