Bias and variance estimation with boostrap

Question

The Wikipedia article about Jacknife estimation of the bias and variance of an estimator $\theta$ includes the following formulas:

Variance of $\theta$:

$ \operatorname {Var}(\theta )=\sigma ^{2}={\frac {n-1}{n}}\sum _{{i=1}}^{n}({\bar {\theta }}_{i}-{\bar {\theta }}_{{\mathrm {Jack}}})^{2}$

where ${\bar {\theta }}_{{Jack}}={\frac {1}{n}}\sum _{{i=1}}^{n}({\bar {\theta }}_{i})$ is the jacknife estimator.

Bias-correction of $\theta$:

$ {\bar {\theta }}_{{\mathrm {BiasCorrected}}}=N{\bar {\theta }}-(N-1){\bar {\theta }}_{{Jack}} $

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My question is: What are the corresponding formulas for bootstrap? Are they different?

Answer 1

I think the formulas for Bootstrap are as follows, although I can't seem to find a proper reference:

Let $h = \frac{1}{K} \sum_{k=1}^{K}{\hat{y}_k}$ be the mean predicted outcome from the $K$ bootstrap resamples.

Then:

• Bias is $y - h$
• Variance is $\frac{1}{K-1}\sum_{k=1}^{K} (y_k - h)^2$