Confidence region ($C(X)$)

A random set of parameter values with specified repeated-sampling coverage: ${\mathbb{P}}_{\theta }\{\theta \in C(X)\}\ge 1-\alpha$ for every permitted $\theta$. Exact coverage, asymptotic coverage and pointwise versus uniform validity must be distinguished. A confidence interval is the scalar interval case.

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.