# estimate multivariate normal parameters

Suppose that I have "a realization" of random vector $$x=(x_1,\cdots,x_N)$$ where $$N$$ is sufficiently large $$N>100$$. I know that random vector is joint normally distributed $$x \sim N(\mu,\Sigma),$$ where diagonal elements are $$\sigma^2$$ and off-diagonal elements are $$\rho\sigma^2$$.

I wonder how I could estimate $$\mu,\Sigma$$ when I only observe one sample. I guess $$\mu$$ and $$\sigma^2$$ can be estimated from a sample mean and variance, following the law of large numbers. I wonder if it is possible to estimate $$\rho$$.

• $\mu$ should be written as $\mu\mathbf 1$ to alleviate the confusion Aug 16, 2021 at 12:18

With only one sample, the maximum likelihood estimate is $$\mu=x, \Sigma=0$$.
• Yes, I understand that I cannot do much with one sample in general. But here, I assume that $\mathbb{E}x_i = \mu$ for all $i$, and $var(x_i) = \sigma^2$ for all $i$, and $cov(x_i,x_j) = \rho \sigma^2$ for all $i\ne j$. By this, I said $\mu$ can be estimated by sample mean and law of large number. Sorry that I was not clear enough. Aug 16, 2021 at 6:45