Estimating Extreme Quantiles with the Fisher Approximation of the Tail

Overview

Estimating a very high quantile means learning about a region where observations are scarce. At the 99.9th percentile, only one observation in a thousand is expected to lie beyond the target. A tail model helps us use the available extreme observations to estimate these rare-event quantities.

The Peaks over Threshold approach starts by choosing a high threshold \(u\). We retain observations above it and model their excess \(Y=X-u\), conditional on \(X>u\). If \(F\) is the distribution of the original observations, the excess distribution is

Extreme value theory motivates the Generalized Pareto Distribution (GPD) as an approximation to this conditional tail at sufficiently high thresholds, under suitable assumptions on \(F\). Raising the threshold can improve the approximation, while leaving fewer observations to fit it. A practical threshold therefore balances approximation bias against estimation uncertainty.

This work focuses on the heavy-tailed case, with positive shape \(\xi\) and scale \(\sigma\). The GPD survival function gives the probability that an excess is larger than \(y\geq 0\):

The thesis explores a more flexible approximation for the finite thresholds we can actually use. We fit a scaled Fisher distribution, equivalently a beta-prime distribution, to the excesses. It has two shape parameters \(a,b>0\) and a scale \(s>0\). For positive excesses \(y>0\), its density is

Here \(B(a,b)\) is the beta function. The parameter \(a\) changes the shape close to the threshold, while \(b\) controls the far-tail decay: \(f(y)\) is proportional to \(y^{-b-1}\) asymptotically. This separates the behavior of the observed excesses near zero from the rate at which the distant tail decays.

The GPD sits inside this family. Setting

recovers the positive-shape GPD exactly. Allowing \(a\) to vary adds one degree of freedom around that familiar baseline. The aim is to better describe excesses at a usable threshold while retaining a power-law tail for extrapolation.

The project considers maximum likelihood, probability-weighted moments, and the method of moments, alongside Bayesian inference. The Bayesian treatment uses Jeffreys priors, constructed from the Fisher information, and MCMC to sample the posterior. This lets uncertainty in the fitted parameters carry through to the estimated quantiles.

Once the excess distribution \(H\) is fitted, we reconnect it to the original observations. If \(k\) out of \(n\) observations exceed \(u\), then the exceedance probability is estimated by \(p_u\approx k/n\). For a target probability \(p>F(u)\), the original-scale quantile is estimated using

This expression makes the comparison concrete: the fitted tail determines how far beyond the threshold we must go to reach the desired quantile. The reported experiments compare fitted and true quantiles as the number of excesses changes, illustrating improved estimates with the Fisher model in the examples studied, particularly with limited tail data.

The extra flexibility also needs to be estimated from the available observations. Its usefulness depends on the threshold, the tail sample size, and the underlying distribution. The central idea is to adapt a theoretically motivated tail model to the finite-data regime in which extreme quantiles are estimated.

Thesis context

This project came from my MSc work on extreme-value statistics. The research was carried out at Inria Grenoble with Julyan Arbel during my Erasmus exchange. The MSc degree was granted by the University of Warsaw, under the supervision of John M. Noble at the Faculty of Mathematics, Informatics and Mechanics. All errors are my own.

@mastersthesis{lewandowski2018fisher,
  title = {Estimating Extreme Quantiles with the Fisher Approximation of the Tail},
  author = {Lewandowski, Michal},
  school = {University of Warsaw},
  year = {2018},
  note = {MSc thesis research carried out at Inria Grenoble with Julyan Arbel during an Erasmus exchange; supervised at MIM UW by John M. Noble}
}