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In a classic paper on conducting bootstrapped hypothesis tests, Hall & Wilson (1991) present the following guideline, which I am trying to understand:

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On first read, this seemed obvious: $\widehat{\theta}^{*}$ are being resampled under $H_0$, so are consistent for $\theta_0$, whereas $\widehat{\theta}$ is consistent for $\theta$. Thus, $\widehat{\theta}^{*} - \widehat{\theta}$ is consistent for $\theta_0 - \theta$, which makes sense. In contrast, $\widehat{\theta}^{*} - \theta_0$ is consistent for $\theta_0 - \theta_0 = 0$, which is not useful.

But then I realized, on rereading the paragraph above the First Guideline and a subsequent example, that Hall & Wilson are computing $\widehat{\theta}^{*}$ by resampling directly from the sample, rather than under the null. In that case, I have no idea why they recommend resampling $\widehat{\theta}^{*} - \widehat{\theta}$, which is consistent for 0!

References

Hall, P., & Wilson, S. R. (1991). Two guidelines for bootstrap hypothesis testing. Biometrics, 757-762.

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The idea is that irrespective of the choice of $\theta_0$ the bootstrap distribution of $\widehat\theta^*$ - $\widehat\theta$ resembles the sampling distribution of $\widehat\theta$ - $\theta$. If you instead use $\theta_0$ you will be pulling the estimate closer to $\theta_0$ and thus lose power under the alternative. I think this is essentially what Hall and Wilson are saying but possibly a little simpler and hopefully clearer for you.

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  • $\begingroup$ Thanks -- I think I understand. Is it correct to say that this algorithm allows us to conduct a hypothesis test under $H_0$ while avoiding the difficulties of resampling under $H_0$ (because we're just resampling from the original)? $\endgroup$
    – half-pass
    Jul 7, 2017 at 2:42
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    $\begingroup$ That sounds like a reasonable statement to me. $\endgroup$
    – Michael R. Chernick
    Jul 7, 2017 at 2:44
  • $\begingroup$ Interesting. (As an aside, your textbook on bootstrapping is wonderful -- thank you!) $\endgroup$
    – half-pass
    Jul 7, 2017 at 2:56
  • $\begingroup$ Thanks for the compliment. I briefly discuss the Hall-Wilson paper in my book. You may also like to look at Peter Hall's book which also covers this hypothesis testing issue. $\endgroup$
    – Michael R. Chernick
    Jul 7, 2017 at 3:04

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