Beyond the sample: extreme quantile and probability estimation

Economic problems such as large claims analysis in insurance and value-at-risk in fi- nance, require assessment of the probability P of extreme realizations Q. This paper provides a semi-parametric method for estimation of extreme (P, Q) combinations for data with heavy tails. We solve the long standing problem of estimating the sample threshold of where the tail of the distribution starts. This is accomplished by the combination of a control variate type device and a subsample bootstrap technique. The sub- sample bootstrap attains convergence in probability, whereas the full sample bootstrap would only provide convergence in distribution. This permits a complete and comprehensive treatment of extreme (P, Q) estimation.