Hypothesis Test for mean vector of Multivariate Normal Distribution Given two independently $X_1$ and $X_2$ and that these are bivariately normally distributed with mean vector components $\mu_1$ and $\mu_2$ and variance-covariance matrix shown below:
If we want to test $H_0: \mu_1=0$ v.s. $H_a: \mu_1\neq 0$. How to show that the likelihood ratio test for testing $H_0$?

A likelihood ratio statistic is equal to:
$$ \mathcal{L}(\hat{\mu}_{M}) - \mathcal{L}(\hat{\mu}_{constr}) $$
where $\mathcal{L}(\cdot)$ is the normal log-likelihood function, $\hat{\mu}_{M}$ is the maximum-likelihood estimator, and
$$ \hat{\mu}_{constr} = {\arg\max}_{\{\mu \in \mathbb{R}^2: \ \mu_1\mu_2 = 0 \}} \mathcal{L}(\mu) $$
You can easily solve for the constrained estimator by computing

*

*The $\mu_1$ maximizer of $\mathcal{L}(\mu)$ given $\mu_2 = 0$.

*The $\mu_2$ maximizer of $\mathcal{L}(\mu)$ given $\mu_1 = 0$.

*Choose the solution with the largest likelihood.

