I would like to know how to derive the asymptotic covariance matrix of the maximum likelihood estimator of the two parameters (mean vector and covariance matrix) of a multivariate normal distribution. While there are many online sources explaining how to derive the estimators (e.g., Maximum Likelihood Estimators - Multivariate Gaussian), I was not able to find a derivation (not even a formula) for their asymptotic covariance matrix (the matrix should also contain the covariance between the estimator of the mean vector and the estimator of the covariance matrix, which I guess is zero). I tried to derive it myself but I am stuck because probably something needs to be vectorized but I do not know how to do it exactly.


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