# 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 $$H_0$$?

• Please add the self-study tag and read its wiki. Jan 15, 2022 at 6:32
• 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. Jan 16, 2022 at 6:27

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 $$\mu_1 = 0$$.