# Why resample by centering vs. the sample estimate, not null parameter value?

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

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.

## 1 Answer

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.

• 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)? Jul 7 '17 at 2:42
• That sounds like a reasonable statement to me. Jul 7 '17 at 2:44
• Interesting. (As an aside, your textbook on bootstrapping is wonderful -- thank you!) Jul 7 '17 at 2:56
• 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. Jul 7 '17 at 3:04