Question

Use the problem of estimating the mean of a standard Cauchy distribution to illustrate how the bootstrap can fail for heavy-tailed distributions.

Answer 1

Answer

We know that for Cauchy distribution the variance is infinite. That is, for this case true population variance is infinite. Validity of bootstrap method depends on the sampled data which is approximately distributed as the true population (NOT exactly same). For any sample, variance will be finite. But for Cauchy distribution variance is infinite. So this is a big problem for estimating mean of a standard Cauchy distribution. So in practice, we don't think of sampling data from a population with an infinite variance. Similarly for heavy tailed distributions, population variance is not finite. So bootstrap method fails for heavy tailed distribution.