Beyond the Generalized Pareto: A Fisher / Beta-Prime Tail Model for Peaks over Threshold

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.

Problem Statement

Extreme Value Theory gives a principled way to estimate rare-event quantities, but the hardest practical cases are finite. If \(X_1,\ldots,X_n\) are observations from an unknown distribution \(F\), the target may be a high quantile

\[ q_p = F^{-1}(p), \qquad p \approx 0.99,\;0.995,\;0.999. \]

At those levels, the empirical sample contains limited direct information. The Peaks over Threshold method therefore chooses a high threshold \(u\), keeps only exceedances \(Y=X-u\mid X>u\), and models their conditional distribution.

The standard model

The classical exceedance model is the Generalized Pareto Distribution:

\[ G_{\xi,\sigma}(y) = 1-\left(1+\frac{\xi y}{\sigma}\right)^{-1/\xi}, \qquad 1+\frac{\xi y}{\sigma}>0. \]

The GPD is justified asymptotically as the threshold moves into the tail. In real datasets, however, the threshold cannot be pushed arbitrarily high: a higher threshold reduces bias but leaves fewer exceedances. The thesis targets this middle regime: finite thresholds, small tail samples, and extreme quantile targets.

The Fisher / beta-prime extension

The proposed alternative is a scaled beta-prime distribution, also known as a Pearson Type VI or Fisher-type model:

\[ f(y\mid a,b,s) = \frac{1}{sB(a,b)} \left(\frac{y}{s}\right)^{a-1} \left(1+\frac{y}{s}\right)^{-a-b}, \qquad y>0. \]

The parameter \(b\) controls the far-tail decay, \(s\) controls scale, and \(a\) changes the curvature close to the threshold. That extra near-threshold flexibility is the reason the model matters as a finite-threshold correction.

Why it is not arbitrary

The beta-prime model contains the positive-shape GPD as a special case. Use

\[ a=e^\delta,\qquad b=\frac{1}{\xi},\qquad s=\frac{\sigma}{\xi},\qquad \xi>0. \]

When \(\delta=0\), we have \(a=1\), and the beta-prime density becomes the positive-shape GPD density:

\[ f(y) = \frac{1}{\sigma} \left(1+\frac{\xi y}{\sigma}\right)^{-1-1/\xi}. \]

This makes the model a nested extension rather than a disconnected alternative. The extra parameter \(\delta\) measures departure from the GPD baseline.

Quantile estimation

If \(p_u=P(X>u)\) and \(H\) is the fitted exceedance distribution, then

\[ P(X>u+y)=p_u\{1-H(y)\}. \]

Solving for the original-scale quantile gives

\[ q_p = u + H^{-1}\left(1-\frac{1-p}{p_u}\right). \]

This equation isolates the model comparison. With the threshold and exceedance rate fixed, the difference between the GPD and beta-prime approaches is the fitted conditional quantile \(H^{-1}\).

Modern framing

The most conservative framing is not to replace the GPD, but to use a nested Fisher / beta-prime exceedance model as a finite-threshold correction for extreme quantile estimation.

A Bayesian extension would place shrinkage on the extra parameter, for example

\[ \delta\sim N(0,s_\delta^2). \]

This pulls the model back toward the GPD when the exceedance data do not support additional flexibility. The right evaluation target is not only likelihood, but extreme-quantile accuracy, threshold stability, and robustness when the ordinary GPD is already adequate.

@mastersthesis{lewandowski2018fisher,
  title = {Beyond the Generalized Pareto: A Fisher / Beta-Prime Tail Model for Peaks over Threshold},
  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}
}