# How should bootstrap bias correction be applied when combining multiple biased estimates?

I am trying to use non-parametric bootstrap methods to get confidence intervals for a multi-step estimator that combines multiple variables:

$$\hat\theta = \sqrt{\sum\limits_{i=1}^{n} {\left(\sum\limits_{t=1}^{m}|\overline{X_{i,t}} - \overline{Y_{i,t}}|\right)}^2}$$

I find a bootstrapped distribution $$\hat\theta^\ast_b$$ by resampling each $$X_{i,t}$$ and $$Y_{i,t}$$ independently 10,000 times.

Since $$|\overline{X_{i,t}} - \overline{Y_{i,t}}|$$ is already a biased estimator, the bootstrap distributions at this stage (orange in the figure below) are already systematically positively biased relative to the estimated $$|\overline{X_{i,t}} - \overline{Y_{i,t}}|$$ (blue line in the figure below), especially when $$|\overline{X_{i,t}} - \overline{Y_{i,t}}|$$ is small:

The vector norm operation then compounds these biases. The result is that $$\hat\theta$$ (green line in the plot below) sometimes falls completely outside of the bootstrapped distribution $$\hat\theta^\ast_b$$ (red in the plot below).

I understand that BCa (bias corrected and accelerated) bootstrap is a preferred method for generating confidence intervals in the presence of bias and skew. However, this is problematic here, since $$z_0 = -\inf$$ when all $$\hat\theta^\ast_b > \hat\theta$$.

Thus my questions are:

Would it be valid to use bootstrap bias estimation to adjust for estimated bias by reporting $$\tilde\theta$$, where $$\tilde\theta = 2\times{\hat\theta} - \bar{\hat\theta^\ast}$$, and reporting percentile confidence intervals by re-centering $$\hat\theta^\ast_b$$ around $$\tilde\theta$$?

Should such bias correction be applied to each $$|\overline{X_{i,t}} - \overline{Y_{i,t}}|$$ before combining them?

If all I am interested in is relative values of $$\hat\theta$$ under different conditions, should I just report the means of the bootstrapped distributions, $$\overline{\hat\theta^\ast}$$ with percentile confidence intervals?

Is it even valid to use non-parametric bootstrap methods in this case?

Thanks very much for your consideration.

• Have you tried the BCa bootstrap in any event? This problem seems very similar to the one on this page, for the intrinsically biased statistic of Shannon entropy. Sometimes you need to do some transformation on your statistic to get it closer to a pivotal quantity. Sometimes nothing works. See discussion and references on this page. Please look at those discussions, see if they help, and perhaps then edit this question to focus on any yet-unresolved problems.
– EdM
Aug 20 '20 at 23:55
• Thanks for those links. I am unclear on how to implement the BCa in this case, since the $z_0$ parameter relies on the inverse normal of the number of bootstrapped estimates that are less than the empirical estimate, which in this case is 0, resulting in $z_0=-\inf$. It seems one solution from the Shannon Entropy discussion is to apply a bias adjustment term in the calculation of the entropy itself. I am trying to figure out if there is such an adjustment I can apply during the absolute difference step and the vector norm step in my formula. Aug 22 '20 at 1:48