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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 $H_0$?

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  • $\begingroup$ Please add the self-study tag and read its wiki. $\endgroup$ Jan 15, 2022 at 6:32
  • $\begingroup$ You need to revise your solution to the constrained optimization problem. You can satisfy the constraint if $\mu_1 = 0$ or $\mu_2 = 0$ (or both). You need to solve each problem independently (setting one of the parameters to zero). The solutions are of the form $(\mu_{1,constr},0)$ or $(0,\mu_{2,constr})$. Your current solutions don't look like this. After you have figured the solution to the two simpler problems, you need to figure out which solution has the largest log-likelihood. $\endgroup$ Jan 16, 2022 at 6:27

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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

  1. The $\mu_1$ maximizer of $\mathcal{L}(\mu)$ given $\mu_2 = 0$.
  2. The $\mu_2$ maximizer of $\mathcal{L}(\mu)$ given $\mu_1 = 0$.
  3. Choose the solution with the largest likelihood.
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