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 $\rho$.

With only one sample, the maximum likelihood estimate is $\mu=x, \Sigma=0$.
In general, this problem is a estimation problem for an exponential family.  Take the expected value of the sufficient statistics of your samples—these are the expectation parameters of the maximum likelihood distribution.
