Confidence region

In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an n-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, although other shapes can occur.

The confidence region is calculated in such a way that if a set of measurements were repeated many times and a confidence region calculated in the same way on each set of measurements, then a certain percentage of the time (e.g. 95%) the confidence region would include the point representing the "true" values of the set of variables being estimated. However, unless certain assumptions about prior probabilities are made, it does not mean, when one confidence region has been calculated, that there is a 95% probability that the "true" values lie inside the region, since we do not assume any particular probability distribution of the "true" values and we may or may not have other information about where they are likely to lie.

Suppose we have found a solution ${\displaystyle {\boldsymbol {\beta }}}$ to the following overdetermined problem:

where Y is an n-dimensional column vector containing observed values of the dependent variable, X is an n-by-p matrix of observed values of independent variables (which can represent a physical model) which is assumed to be known exactly, ${\displaystyle {\boldsymbol {\beta }}}$ is a column vector containing the p parameters which are to be estimated, and ${\displaystyle {\boldsymbol {\varepsilon }}}$ is an n-dimensional column vector of errors which are assumed to be independently distributed with normal distributions with zero mean and each having the same unknown variance ${\displaystyle \sigma ^{2}}$.

...
Wikipedia