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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 $\rho$.

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    $\begingroup$ $\mu$ should be written as $\mu\mathbf 1$ to alleviate the confusion $\endgroup$
    – Xi'an
    Aug 16, 2021 at 12:18

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user1292919
    Aug 16, 2021 at 6:45
  • $\begingroup$ @user1292919 Oh! I thought you were considering a curved exponential family. I'm not sure then. I'd have to think about it more. $\endgroup$
    – Neil G
    Aug 16, 2021 at 14:45

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