Maximum-likelihood Estimation For The Multivariate Normal Distribution
A random vector X ∈ Rp (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ Rp × p is a positive-definite matrix and the probability density function of X is
where μ ∈ Rp×1 is the expected value of X. The covariance matrix Σ is the multidimensional analog of what in one dimension would be the variance, and normalizes the density so that it integrates to 1.
Suppose now that X1, ..., Xn are independent and identically distributed samples from the distribution above. Based on the observed values x1, ..., xn of this sample, we wish to estimate Σ.
Read more about this topic: Estimation Of Covariance Matrices
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