Some Criticisms of Stochastic Shrinkage and Ridge Regression, With Counterexamples

The existence theorems for reduced mean squared error ridge regression and shrunken estimators are commonly used to justify the use of completely stochastic versions of these same estimators. But there are no guarantees for improvement over the original unbiased estimators. In particular, this is demonstrated for a case where all possible values of the stochastic factor are known to satisfy the relevant inequality of the corresponding existence theorem. Examples of estimators with guaranteed improvement are given.