ASYMPTOTIC SIZE AND A PROBLEM WITH SUBSAMPLING AND WITH THE m OUT OF n BOOTSTRAP

This paper considers inference based on a test statistic that has a limit distribution that is discontinuous in a parameter. The paper show; that subsampling and m out of n bootstrap tests based on such a test statistic often have asymptotic size-defined as the limit of exact sire-that is greater than the non-final level of the tests. This is due to a lack of uniformity in the pointwise asymptotics. We determine precisely the asymptotic size of such tests under a general set of high-level conditions that are relatively easy to verify. The results show that the asymptotic size of subsampling and m out Of n bootstrap tests is distorted in some examples but not in others.