Bayesian Estimation of Extreme Quantiles and the Exceedance Distribution for Paretian Tails

Abstract

Estimating extreme quantiles is an important task in many applications, including financial risk management and climatology. More important than estimating the quantile itself is to insure zero coverage error, which implies the quantile estimate should, on average, reflect the desired probability of exceedance. In this research, we show that for unconditional distributions isomorphic to the exponential, a Bayesian quantile estimate results in zero coverage error. This compares to the traditional maximum likelihood method, where the coverage error can be significant under small sample sizes even though the quantile estimate is unbiased. More generally, we prove a sufficient condition for an unbiased quantile estimator to result in coverage error. Interestingly, our results hold by virtue of using a Jeffreys prior for the unknown parameters and is independent of the true prior. We also derive an expression for the distribution, and moments, of future exceedances which is vital for risk assessment. We extend our results to the conditional tail of distributions with asymptotic Paretian tails and, in particular, those in the Fr\'echet maximum domain of attraction. We illustrate our results using simulations for a variety of light and heavy-tailed distributions.