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Abstract:
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In this paper, we first prove that some inconsistent uniform bootstrap distribution estimators, viewed as random elements in the space of distributions on the real line, converge in distribution to some random distributions. Second, we define the hybrid biased-bootstrap, a consistent semiparametric bootstrap procedure obtained by re-sampling from a weighted empirical distribution on the sample corresponding to an appropriately chosen estimate of the parameter. We illustrate our both theoretical and empirical results on some classical examples where the naive bootstrap fails, such as super-efficient estimators, the non-negative mean, and the squared mean. Simulations demonstrate the consistency of this approach, but also show that the inconsistent uniform bootstrap can be better with respect to minimax risk, echoing similar findings in the literature for the m-out-of-n bootstrap.
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