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A Simple Counterexample to the Bootstrap

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Abstract

The bootstrap of the maximum likelihood estimator of the mean of a sample of iid normal random variables with mean mu and variance one is not asymptotically correct to first order when the mean is restricted to be nonnegative. The problem occurs when the true value of the mean mu equals zero. This counterexample to the bootstrap generalizes to a wide variety of estimation problems in which the true parameter may be on the boundary of the parameter space. We provide some alternatives to the bootstrap that are asymptotically correct to first order. We consider two types of bootstrap percentile confidence intervals in the above example. We find that they both have asymptotic coverage probability that exceeds the nominal asymptotic level when the true value of the mean it equals zero.

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  • Donald W.K. Andrews, 1997. "A Simple Counterexample to the Bootstrap," Cowles Foundation Discussion Papers 1157, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1157
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d11/d1157.pdf
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    Cited by:

    1. Cowell, Frank & Victoria-Feser, Maria-Pia, 1998. "Statistical inference for Lorenz curves with censored data," LSE Research Online Documents on Economics 2049, London School of Economics and Political Science, LSE Library.
    2. Chunlin Wang & Paul Marriott & Pengfei Li, 2022. "A note on the coverage behaviour of bootstrap percentile confidence intervals for constrained parameters," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 809-831, October.

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