Post-Selection Inference

Abstract

Researchers are often interested in making inference on one or a few "best" treatment out of p given treatments. This problem is referred to as post-selection inference. Substantial research has been done on constructing point estimates and confidence sets for a multivariate normal mean vector, but there are very few works on inference after selection. Often, out of all available estimates, the ones that are minimax and/or admissible are preferred. It was shown by Sackrowitz & Samuel-Cahn (1986) that X1 , the first order statistic, is minimax for estimating the selected mean for p [LESS-THAN OR EQUAL TO] 3, but it is not minimax for p > 3, but the question whether it is admissible was still open. In Chapter 1, following the arguments of Berger (1976a) and Maruyama (2009) we prove that X1 is admissible for p < 4 and some bias correction is needed for p [GREATER-THAN OR EQUAL TO] 4. The bias corrected estimate, a generalized Bayes estimate, will be admissible for p [GREATER-THAN OR EQUAL TO] 4. We also provide a comparison of this admissible estimator under the horseshoe prior and other estimators proposed in the literature, including the estimator proposed in Reid & Tibshirani (2014) and a generalized Bayes estimator under the horseshoe prior. Naive confidence sets under selection fail to maintain the nominal coverage probability. In Chapter 2, by approximating the coverage probability of the naive set, we derive two confidence sets that maintain coverage probability. The first set is straightforward to construct, but it results in a large conservative set with coverage probabilities close to one. The second set is more involved to construct, but it is significantly improve over the first one. In Chapter 3 we explore minimaxity and admissibility of the naive set under selection. If we require a set to maintain coverage probability, we show that it is only minimax for p = 2. Almost admissibility is shown for p = 2. The rest of the chapter is dedicated to investigating potential dominating sets.