I am trying to understand a fundamental property of the Likelihood Ratio Test (LRT).

For simplicity, the following example is framed as binomial data, which of course could be solved using a simple hypothesis test (the binomial test). But for the sake of the argument, please consider it from the LRT perspective, since it is a nice simplification of more complicated scenarios.

Let $X$ ~ binomial$(n,p)$. Let $p_0 = 0.2$.

I would like to test that $H_0: p \in (0 , p_0)$ vs. $H_1: p \in (p_0 , 1)$.

Then my LRT test statistic is, as usual:

$\Lambda = - 2 \cdot \log \frac{sup \{L(\theta | x) : \theta \in (0 , p_0) \}}{sup \{L(\theta | x) : \theta \in (0,1)\}} $

I have read that the likelihood ratio test is not universal. I've read that it fails when:

1. the parameter is constrained to be on the boundary (e.g. $N_2( [\mu_1 , \mu_2 ] , \Sigma)$ where $\mu_1, \mu_2 \ge 0$ and $H_0: \mu_1 = \mu_2 = 0$ ).
2. The parameters must be on the "interior" of the parameter space.

In these cases, the LRT statistic $\Lambda$ is not asympotitcally $\chi^2$ distributed.

In the topological sense, my $H_0$ constrained parameter space $(0,p_0)$ is indeed an interior set to $(0,1)$. Further, I am not constraining $p$ to be on the boundary of $(0,1)$. Seems like I should be OK, right?

Is the LRT statistic above still valid ? i.e. is $\Lambda$ above still asymptotically ~ $\chi^2$ ?

Lastly: what is the degrees of Freedom of my LRT test?

I am trying to understand a fundamental property of the Likelihood Ratio Test (LRT).

For simplicity, the following example is framed as binomial data, which of course could be solved using a simple hypothesis test (the binomial test). But for the sake of the argument, please consider it from the LRT perspective, since it is a nice simplification of more complicated scenarios.

Let $X$ ~ binomial$(n,p)$. Let $p_0 = 0.2$.

I would like to test that $H_0: p \in (0 , p_0)$ vs. $H_1: p \in (p_0 , 1)$.

Then my LRT test statistic is, as usual:

$\Lambda = - 2 \cdot \log \frac{sup \{L(\theta | x) : \theta \in (0 , p_0) \}}{sup \{L(\theta | x) : \theta \in (0,1)\}} $

I have read that the likelihood ratio test is not universal. I've read that it fails when:

1. the parameter is constrained to be on the boundary (e.g. $N_2( [\mu_1 , \mu_2 ] , \Sigma)$ where $\mu_1, \mu_2 \ge 0$ and $H_0: \mu_1 = \mu_2 = 0$ ).
2. The parameters must be on the "interior" of the parameter space.

In these cases, the LRT statistic $\Lambda$ is not asympotitcally $\chi^2$ distributed.

In the topological sense, my $H_0$ constrained parameter space $(0,p_0)$ is indeed an interior set to $(0,1)$. Further, I am not constraining $p$ to be on the boundary of $(0,1)$. Seems like I should be OK, right?

Is the LRT statistic above still valid ? i.e. is $\Lambda$ above still asymptotically ~ $\chi^2$ ?

Based on my reasoning about the interior point/constraints above, I believe the answer is yes.

Lastly: what is the degrees of Freedom of my LRT test?

I believe it is 1, intuitively.