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 estimato r to the usual F confidence set for the p regression parameters of a l inear model. Previous studies usually focused on the known-variance ca se and typically concluded that whatever holds in the known-variance c ase also holds in the unknown-variance case when the variance is repla ced by its best linear estimator based on S2, where S2 is the sum of s quared residuals. We are surprised that this is not entirely the scena rio we observe here. In fact, in many situations involving unknown var iance, the range of the shrinkage factor a for the associated confiden ce set to have uniformly higher coverage probabilities than its F coun terpart can be ten times bigger than 2(p - 2) (the expected upper boun d in the known-variance case). This is true especially when the degree s of freedom are small. Our numerical studies also show that to gain s ubstantial 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.