Is bootstrap Studentized pivotal interval always better ?

I am trying to do exercice 6 of chapter 3 in "All of non parametric statistics" by Wasserman which consists in evaluating the 4 type of bootstrap intervals (normal, percentile, pivotal, studentized) for log-normal samples and the estimator of skewness.

what I did :

• For a given n, I draw normal samples Y_i, take X_i = exp(Y_i)
• Run bootstrap (by drawing a sample of size n with replacement from the original data X)
• Compute the four 95% confidence intervals. For the studentized interval, as suggested in the book, I estimated the standard error for each bootstrap replication using the non-parametric delta method.
• For each intervals, check if the true value of skewness is inside the intervals or not. I do that a lot of times and count how often the interval was good or not. When n->infinity, the proportion should tend to 95%.

For the log-normal distribution, here are the evolution of this proportion when n is increasing

The four interval are not very good but the studentized interval is the better.

I then tried to do the same with normal distribution. And here is what I get

The studentized CI seems to be the less good (and it does not seem to increase).

So my questions are :

• Is this behaviour normal/possible ?
• Do you think that it more probably means that I made some error in my simulations ? (I can naturally show the code if somebody is interested)