I need to deal with the bootstrap-t confidence intervals but I got many doubts about it. I’m not a native

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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 Boot- strap(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 (x1, ..., xn) is a fixed data set from an unknown distribution F , θ is a parameter, not necessary the mean, and θˆ = s(X1, ..., Xn)

is an estimator. We denote with x1∗, ..., xB∗ B bootstrap samples and respec- tively  with  θˆ∗(1), ..., θˆ∗(B)  the  bootstrap  replications.  Following  pages  160-161

of Efron’s book, it uses the following estimator to find bootstrap-t intervals:

θˆ∗(b)−θˆ ∗ ˆ∗

Zb∗ = , where sˆe (b) is the standard error of θ (b). Now I got some

questions to ask:

1. Why he took exactly Zb∗  and not, for example, just θˆ∗(b) − θˆ?  In fact, we can’t say that Zb∗ 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 θ.  So we could find αth percentiles that Efron finds in the same way with θˆ∗(b) − θˆ.

2. Why the intervals we found are for θ, when in Zb∗ the parameter θ does’t appear?

3. Why in Zb∗ we use sˆe∗ instead of sˆe?

4. the term θˆin Zb∗ is for s(X1, ..., Xn), so a random variable, or for s(x1, ..., xn), so a fixed value?

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

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