Confidence ellipsoid ($C=\{\theta :(\theta -\hat{\theta }{)}^{T}A(\theta -\hat{\theta })\le c\}$)

An ellipsoidal confidence region defined by a positive-definite matrix $A$ and a calibrated threshold $c$. For a known-covariance multivariate normal distribution sample, $A=n{\Sigma }^{-1}$ and the $1-\alpha$ quantile of the chi-squared distribution with $d$ degrees of freedom give exact coverage for the $d$-dimensional mean.