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?

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.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.