Comment on “The Effect of Model Selection on Confidence Regions and Prediction Regions” by P. Kabaila

In this comment, I first want to offer a few general remarks on the problems caused by nonuniformity in the convergence of coverage probabilities as pointed out in Kabaila's (1995) very stimulating paper. Second, I also argue that within the context of set estimation after model selection one typically will be more interested in conditional coverage probabilities than in unconditional ones. Concentrating on the last example of Section 2 in Kabaila, I then illustrate how a confidence region can be constructed such that the conditional coverage probabilities do not suffer from the nonuniformity problem discussed in Kabaila. As a by-product, I show that the confidence region considered in this example in Kabaila is not a very natural one and that it has undesirable conditional coverage properties, which I find actually more troublesome than the lack of uniformity in the unconditional coverage probability pointed out by Kabaila.