A confidence bound approach to choosing the biasing parameter in ridge regression

Consider the multiple linear regression model Y = Xβ + ∈, ∈ ∼ Ň(0, σ²I) where the matrix S = X’ X is ill conditioned. A confidence bound approach is developed for choosing k in the ridge estimator β*(k) = (S + kI)^–1X’ Y as follows: A parameter (Equation presented) is defined that is essentially the largest (constant) k one could use and still have β*(k)’s mean squared error (MSE) be less than (Equation presented) MSE (where (Equation presented) is the usual estimate of β). A procedure is then developed to obtain a lower confidence bound k_γfor (Equation presented), and the estimator β* ≡ β*(k_γ) is considered.