Generalized Pareto Distribution

Statement¶

The generalized Pareto distribution is the standard limit family for threshold excesses. With location zero, an excess variable $Y\ge0$ has shape $\xi$ and scale $\beta>0$ when

for $y\ge0$ with $1+\xi y/\beta>0$. For $y<0$ the CDF is zero; for $\xi<0$ it is one at and above $-\beta/\xi$, by continuity. The boundary case is

For $\xi\ge0$, the support is $y\ge0$. For $\xi<0$, the support is $0\le y\le -\beta/\xi$, so the tail has a finite endpoint. The survival function for $\xi\ne0$ is

The mean exists only for $\xi<1$ and equals $\beta/(1-\xi)$. The variance exists only for $\xi<1/2$ and equals

Shape and endpoint intuition¶

The generalized Pareto distribution is the peaks-over-threshold counterpart of the generalized extreme-value distribution for block maxima. Once a threshold is high enough, the distribution of the excess over that threshold is modeled with one shape parameter $\xi$ and one scale parameter $\beta$.

The sign and size of $\xi$ carry the tail story. Positive $\xi$ gives a Pareto-type heavy tail. Zero gives the exponential boundary. Negative $\xi$ gives a finite endpoint. In the fat-tail examples, the most important boundary values are $\xi=1/2$ for variance and $\xi=1$ for the mean.

For $\beta=1$, the three examples from the survival formula are:

What to notice. Positive $\xi$ gives the slowest-decaying curve, $\xi=0$ is the exponential boundary, and negative $\xi$ ends at the finite endpoint $-\beta/\xi$.

Use generalized_pareto. The endpoint calculation is visible in the returned CDF and survival:

from xmath.incerto.distributions import generalized_pareto

law = generalized_pareto(-0.4, beta=1)
for y in (0, 1, 2.5, 3):
    print(f"y={y:.4g}: pdf={law.pdf(y):.4g}, cdf={law.cdf(y):.4g}, sf={law.sf(y):.4g}")

Output

y=0: pdf=1, cdf=0, sf=1
y=1: pdf=0.4648, cdf=0.7211, sf=0.2789
y=2.5: pdf=0, cdf=1, sf=0
y=3: pdf=0, cdf=1, sf=0

Regenerate the card with uv run python demos/incerto/distribution_cards.py.

Threshold-excess stability and Pareto special case¶

A GPD is stable under raising the threshold. If $Y$ has parameters $\xi,\beta$ and $u\ge0$ satisfies $\beta+\xi u>0$, then, for admissible $y\ge0$ and $\xi\ne0$,

For $\xi=0$, the ratio is $e^{-y/\beta}$. Thus excesses have the same shape and updated scale $\beta+\xi u$. Combining this with the GPD mean gives the mean-excess formula.

The Pareto Type I distribution is an exact generalized Pareto excess model. If $X$ has Pareto tail exponent $\alpha$ and threshold $u\ge x_m$, then

This is the generalized Pareto survival function with

The Pickands-Balkema-de Haan theorem explains why the same family appears asymptotically for a much larger class of threshold exceedances.

Formula check¶

A concrete choice of shape and scale illustrates both the survival and moments.

For $\xi=0.4$ and $\beta=2$, direct substitution gives

Thus $\bar G(0)=1$, $\bar G(1)=1.2^{-2.5}$, $\bar G(5)=2^{-2.5}$, and $\bar G(10)=3^{-2.5}$. These are formula evaluations, with no numerical fitting or simulation.

Caveats¶

• A GPD fit is a tail model for exceedances, not a model for the full distribution body.

• The threshold $u$ is not chosen by the theorem. Shape and scale estimates should be inspected across a range of thresholds.

• When $\xi\ge1$, the fitted excess distribution has no finite mean. When $\xi\ge1/2$, it has no finite variance.

• Dependence, truncation, censoring, seasonality, and mixture effects can distort threshold-excess fits.

References¶

• Balkema and de Haan, “Residual Life Time at Great Age” 1974.

• Pickands, “Statistical Inference Using Extreme Order Statistics” 1975.

• Davison and Smith, “Models for Exceedances over High Thresholds” 1990.

• Coles, An Introduction to Statistical Modeling of Extreme Values 2001.

• Embrechts, Klueppelberg, and Mikosch, Modelling Extremal Events 1997.

Backlinks¶

• Depends on: Extreme-Value Index Estimation and the shared tail notation in Shared notation.

• Used by: Pickands-Balkema-de Haan Theorem, Mean Excess Function, and Tail Threshold Selection.