= Singular covariance and nonexistence of a Gaussian maximum likelihood estimate
For independent $p$-variate <multivariate normal> observations with unknown mean and unrestricted <positive-definite> <covariance matrix>, the centered scatter has <rank> at most $n-1$. If it is singular, choose the mean to be the <sample mean> and shrink the <covariance> on a null direction of the scatter. The quadratic <likelihood> term stays fixed while the <determinant> tends to zero, so the density <likelihood> tends to infinity. Thus a nonsingular <covariance> maximum <likelihood> estimate exists only when the centered scatter is positive definite; for a nonsingular <Gaussian> population this occurs almost surely exactly when $n>p$.
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