Approximation of high quantiles from intermediate quantiles

To estimate a high quantile from a sample of iid random variables, a Generalised Pareto (GP) tail approximation is often applied. Theory supports this if in addition to the GP tail limit, a certain rate is assumed for convergence to this limit. To allow estimation of very high quantiles (at probabilities of exceedance below some power of the number of samples, with the power below -1), a relatively high rate is required, which is very restrictive. A natural relaxation of this assumption leads to alternative tail limits and tail models for the approximation of very high quantiles from intermediate quantiles, which can be estimated from data. A stretched quantile is defined as a convenient analytical surrogate for a high quantile, and a Generalised Weibull (GW) family of distribution functions is shown to characterise limits for the logarithms of stretched quantiles in the same way as the GP family characterises the classical extreme value limits by extended regular variation. Existence of such a log-GW limit (as well as existence of a GW limit, which is a special case) implies that certain probability-based approximation errors vanish locally uniformly for stretched quantiles. As a demonstration, a simple high quantile estimator based on a local log-GW tail model is formulated and is shown to be strongly consistent for very high quantiles if a log-GW limit exists. A numerical simulation illustrates the results.