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Abstract:
Asymptotic bootstrap validity is usually understood as consistency of the distribution of a bootstrap statistic, conditional on the data, for the unconditional limit distribution of a statistic of interest. From this perspective, randomness of the limit bootstrap measure is regarded as a failure of the bootstrap. Nevertheless, apart from an unconditional limit distribution, a statistic of interest may possess a host of (random) conditional limit distributions. This allows the understanding of bootstrap validity to be widened, while maintaining the requirement of asymptotic control over the frequency of corrrect inferences. First, we provide conditions for the bootstrap to be asymptotically valid as a tool for conditional inference, in cases where a bootstrap distribution estimates consistently, in a sense weaker than the standard weak convergence in probability, a conditional limit distribution of a statistic. Second, we prove asymptotic bootstrap validity in a more basic, on-average sense, in cases where the unconditional limit distribution of a statistic can be obtained by averaging a (random) limiting bootstrap distribution. As an application, we establish rigorously the validity of xed-regressor bootstrap tests of parameter constancy in linear regression models.