Confidence sets centered at James—Stein estimators: A surprise concerning the unknown-variance case

We compare the confidence set centered at a James—Stein point estimator to the usual F confidence set for the p regression parameters of a linear model. Previous studies usually focused on the known-variance case and typically concluded that whatever holds in the known-variance case also holds in the unknown-variance case when the variance is replaced by its best linear estimator based on S 2, where S 2 is the sum of squarred residuals. We are surprised that this is not entirely the scenario we observe here. In fact, in many situations involving unknown variance, the range of the shrinkage factor a for the associated confidence set to have uniformly higher coverage probabilities than its F counterpart can be ten times bigger than 2 ( p−2) (the expected upper bound in the known-variance case). This is true especially when the degrees of freedom are small. Our numerical studies also show that to gain substantial improvement one has to use a much larger a, especially when the degrees of freedom of the residuals are small. Application to an economic data set is also included.