  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.
