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Let $\bf{X_{1}}\sim N_p(\mu_1,\Sigma_1)$ and $\bf{X_2\sim N_p(\mu_2,\Sigma_2)}$. Find the likelihood ratio test for $$H_0:\mu_1=\mu_2\ vs\ \mu_1\neq \mu_2$$ with know $\bf{\Sigma_1}$ and $\bf{\Sigma_2}$.

I'm lost in how to do this in the multivariate case, most of the books I've seen found no examples just mention the method and distribution that you get.

I should find the joint likelihood $$L=\prod f_{X_1}(X_1)f_{X_2}(X_2)$$ then find the estimators $\mu_1$, $\mu_2$, $\Sigma_1$, $\Sigma_2$ and then find $$\Lambda=\frac{L(\theta_0)}{L(\theta_1)}$$ the ratio of maximum likelihood under $H_0$ and unrestricted?

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I think that's right. I'm assuming $X_1, X_2$ are mutually independent.

Maximizing the unconstrained likelihood is easy. The only part you neglect to cover is that you need a clever way to maximize the constrained likelihood. Mathematically, constrained optimization is done with Lagrange multipliers. It would be easy to write down such a likelihood. Computationally you don't need to worry about it, just write down the null likelihood and optimize the parameters using a general optimization technique.

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