2009a). Incorrect Asymptotic Size of Subsampling Procedures Based on Post- Consistent Model Selection

Subsampling and the m out of n bootstrap have been suggested in the literature as methods for carrying out inference based on post-model selection estimators and shrinkage estimators. In this paper we consider a subsampling con\u85dence interval (CI) that is based on an estimator that can be viewed either as a post-model-selection estimator that employs a consistent model selection procedure or as a super-e ¢ cient estimator. We show that the subsampling CI (of nominal level 1 for any 2 (0; 1)) has asymptotic con\u85dence size (de\u85ned to be the limit of \u85nite-sample size) equal to zero in a very simple regular model. The same result holds for the m out of n bootstrap provided m2=n! 0 and the observations are i.i.d. Similar zero-asymptotic-con\u85dence-size results hold in more complicated models that are covered by the general results given in the paper and for super-e ¢ cient and shrinkage estimators that are not post-model-selection estimators. Based on these results, subsampling and the m out of n bootstrap are not recommended for obtaining inference based on post-consistent model selection or shrinkage estimators