I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need to deal with the bootstrap-$t$ confidence intervals but I got many doubts about it. I'm not a native speaker but I'll try to be as clear as possible.

We're in a situation where $(x_1,\ldots,x_n)$ is a fixed data set from an unknown distribution $F$, $\theta$ is a parameter, not necessary the mean, and $\hat{\theta}=s(X_1,\ldots,X_n)$ is an estimator.
We denote with $\mathbf{x_1}^*,\dots,\mathbf{x_B}^*$ the $B$ bootstrap samples and respectively with $\hat{\theta}^*(1),\ldots,\hat{\theta}^*(B)$ the bootstrap replications.
Following pages 160-161 of Efron's book, we use the following pivotal quantity to find bootstrap-$t$ intervals: $$ Z^*_b={\hat{\theta}^*(b)-\hat{\theta}\over{\hat{se}^*(b)}}, $$ where $\hat{se}^*(b)$ is the standard error of $\hat{\theta}^*(b)$. Now I got some questions to ask:

  1. Why he took exactly $Z^*_b$ and not, for example, just $\hat{\theta}^*(b)-\hat{\theta}$? In fact, we can't say that $Z^*_b$ is normal or Student, we can't even say that it is close to a Student because we don't know the distribution $F$ and we don't know what is $\theta$. So we could find $\alpha$th percentiles that Efron finds in the same way with $\hat{\theta}^*(b)-\hat{\theta}$.

  2. Why the intervals we found are for $\theta$, when in $Z_b^*$ the parameter $\theta$ doesn't appear?

  3. Why in $Z_b^*$ we use $\hat{se}^*$ instead of $\hat{se}$?

  4. the term $\hat{\theta}$ in $Z_b^*$ is for $s(X_1,\ldots,X_n)$, so a random variable, or for $s(x_1,\ldots,x_n)$, so a fixed value?

  5. Which are the analogies with the $t$-Student?

If you can answer to only one question it will be a huge help for me. Thank you.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.