I read from Wikipedia:
... if one performs a naive bootstrap on the sample mean when the underlying population lacks a finite variance (for example, a power law distribution), then the bootstrap distribution will not converge to the same limit as the sample mean. As a result, confidence intervals on the basis of a Monte Carlo simulation of the bootstrap could be misleading. Athreya states that "Unless one is reasonably sure that the underlying distribution is not heavy tailed, one should hesitate to use the naive bootstrap".
Here's what I understand:
a) Distributions with heavy tails may have infinite variance, or mean (Ex: cauchy distribution)
b) Heavy tailed means that there are a few outliers that are very different from the most of the samples. And these outliers have non-negligible impact on the future statistic procedures.
c) Log-normal or exponential distributions have heavy tails
Here's the question:
- Are my understandings (a, b, c) right?
- Can Bootstrapping be used to estimate confidence interval of mean or variance of lognormal, or exponential population?
- Why does Bootstrapping fail in case of heavy tail?
Please do not use statistical equations (ex: expectations...) as I do not have the background to understand them.