and

Abstract: This paper considers inference based on a test statistic that has a limit distribution that is discontinuous in a parameter. The pa-per shows that subsampling and m out of n bootstrap tests based on such a test statistic often have asymptotic size de\u85ned as the limit of exact size that is greater than the nominal 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