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Assume X $\sim$ Np($\mu$;$\Sigma$), where we know that $\Sigma$ is diagonal. Given an independently identically distributed sample of size n, how can I find a Maximum likelihood estimator for $\mu$ and $\Sigma$?

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This is not an answer to your question, but a hint that should make the answer fairly obvious. If $\Sigma$ is diagonal, then the likelihood function factors into a product of $p$ densities (one could infer this either from the fact that uncorrelated jointly normal variables are independent, or by direct calculation), each of which is functionally independent of all the others in terms of the mean and variance parameters. So you are maximizing a function of this form

$$ \prod_{i=1}^{p} f(x_i, \mu_i, \sigma^2_i) $$

where $x_i$ is the $n$-dimensional sample corresponding to element $i$, and $\mu_i$ and $\sigma^2_i$ the mean and variance parameters for the $i^\text{th}$ univariate distribution. Because of functional independence this function is maximized when each factor has been individually maximized, so your question just reduces to a univariate MLE problem.

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    $\begingroup$ +1 It might even be clearer to argue statistically: a diagonal $\Sigma$ implies the $n$ variables are independent. That reduces the problem to $n$ separate univariate problems. $\endgroup$
    – whuber
    Oct 3 '16 at 15:47

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