Approximation of high quantiles from intermediate quantiles

TL;DR

This paper introduces a novel method for estimating extremely high quantiles based on an alternative tail regularity assumption, the log-GW tail limit, which generalizes existing Weibull tail models, with theoretical and simulation validation.

Contribution

It proposes the log-GW tail limit as an alternative to the Generalised Pareto tail, extending the domain of attraction for high quantile estimation.

Findings

Log-GW tail limit generalizes Weibull tail models.

Convergence results for quantile approximation are established.

Simulations demonstrate advantages and limitations of the method.

Abstract

Motivated by applications requiring quantile estimates for very small probabilities of exceedance, this article addresses estimation of high quantiles for probabilities bounded by powers of sample size with exponents below -1. As regularity assumption, an alternative to the Generalised Pareto tail limit is explored for this purpose. Motivation for the alternative regularity assumption is provided, and it is shown to be equivalent to a limit relation for the logarithm of survival function, the log-GW tail limit, which generalises the GW (Generalised Weibull) tail limit, a generalisation of the Weibull tail limit. The domain of attraction is described, and convergence results are presented for quantile approximation and for a simple quantile estimator based on the log-GW tail. Simulations are presented, and advantages and limitations of log-GW-based estimation of high quantiles are indicated.