# Bootstrap based bias correction

Assume we have a probablistic model $$f_{\theta}(x)$$ and try to estimate the parameter $$\theta$$ based on data $$x$$ with some procedure that yields a biased estimator $$E[\hat{\theta}]=\theta + \eta,$$ where $$\eta$$ represents the bias.

As far as I know we can estimate the bias $$\eta$$ via bootstrap, applying some procedure as follows:

1) Run the model in order to obtain $$\hat{\theta}$$.

2) Draw $$B$$ bootstrap samples $$X_1,...,X_B$$ and re-estimate the model $$B$$ times in order to obtain paramter estimates $$\hat{\theta}_1,...,\hat{\theta}_B$$.

3) Estimate the bias via $$\hat{\eta}=\hat{\theta}-(n^{-1}\sum_{i=1}^B\hat{\theta}_i-\hat{\theta})$$.

However, this procedure implies that the estimate of the bias is $$unbiased$$ itself. I am, however confronted with a situation where $$E[\hat{\theta}]=\theta + \eta(\theta),$$ and hence the bias a function of the unkown $$\theta$$ we cannot obtain valid estimates for $$\eta$$ since simply pluggin-in $$\hat{\theta}$$ for $$\theta$$ does not work as we cannot hope to appoximate $$\eta(\theta)$$ with $$\eta(\hat{\theta})$$. Do you have any suggestions how to solve such a problem (at least approximately)?

• I don't see how the bias can be estimated this way, because when re-estimating the model during each bootstrap iteration you use the same biased model. Can you give a reference on this procedure? – Scholar Feb 18 '19 at 17:20

Thus much depends on how far your term $$\eta(\theta)$$ removes the distribution from being pivotal: providing not just a change of location, which would be OK, but also of scale/shape.