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%.
The four interval are not very good but the studentized interval is the better.
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)
Thanks for your help !