I have a statistic $\hat{\theta}$ for the parameter $\theta$. Which may be biased. Assume $\mathbb{E}[(\hat{\theta}-\theta)^2]=\textit{ecm}^2$ and $\mathbb{V}[\hat{\theta}]=\sigma^2$ are known, but the distribution of $\hat{\theta}$ is unknown.

I want to find a confidence interval for $\theta$ using bootstrap.

I found on the internet that usually one can use bootstrap to find $k_\alpha$ such that $P( \frac{(\hat{\theta}-\theta)^2}{\sigma^2} \leq k_\alpha^2) = 1-\alpha$. I understand this is reazonable because it's analog to normal confidence intervals and in that case it's a pivotal statistic.

My question is, Should I use $\textit{ecm}^2$ in the denominator instead of $\sigma^2$? If the estimator is unbiased both options are the same because ($\sigma^2=\textit{ecm}^2$) But my interest is for biased estimators.

Also, What happens if instead of knowing $\sigma^2$ and $\textit{ecm}^2$ I have their bootstrap estimations?


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