Approximate Confidence Regions for Minimax-Linear Estimators

Minimax estimation is based on the idea, that the quadratic risk
function for the estimate β is not minimized over the entire
parameter space R^K, but only over an area B(β) that is restricted
by a priori knowledge. If all restrictions define a convex area,
this area can often be enclosed in an ellipsoid of the form B(β) =
{ β : β' Tβ ≤ r }. The ellipsoid has a larger volume than the
cuboid. Hence, the transition to an ellipsoid as a priori
information represents a weakening, but comes with an easier
mathematical handling. Deriving the linear Minimax estimator we see
that it is biased and non-operationable. Using an approximation of
the non-central χ^2-distribution and prior information on the
variance, we get an operationable solution which is compared with
OLSE with respect to the size of the corresponding confidence
intervals.