Estimation of Parameters
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant. See estimation of covariance matrices.
In short, the probability density function (pdf) of a k-dimensional multivariate normal is
and the ML estimator of the covariance matrix from a sample of n observations is
which is simply the sample covariance matrix. This is a biased estimator whose expectation is
An unbiased sample covariance is
The Fisher information matrix for estimating the parameters of a multivariate normal distribution has a closed form expression. This can be used, for example, to compute the Cramér–Rao bound for parameter estimation in this setting. See Fisher information for more details.
Read more about this topic: Multivariate Normal Distribution
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