Frechet Distribution and Frechet-Type Limits

Statement¶

The Frechet distribution is the positive-shape extreme-value law for normalized maxima with heavy right tails. In the standard one-parameter form with shape $\alpha>0$, its CDF is

For $y>0$, the density is

It is the $\xi>0$ member of the generalized extreme-value distribution. With $\xi=1/\alpha$, location $\mu=1$, and scale $\sigma=\xi$,

The phrase Frechet-type refers to distributions whose normalized maxima converge to this law. In the usual right-tail setting, this is the regularly varying maximum-domain-of-attraction case: a Pareto-type survival tail with exponent $\alpha$ has extreme-value index $\xi=1/\alpha>0$. Here $F$ is the parent CDF, $\bar F=1-F$ its survival function, $X_i$ its iid draws, and $x_m>0$ the lower cutoff in the exact Pareto example. $\Phi_\alpha$ and $\phi_\alpha$ denote the Frechet CDF and density.

Maxima interpretation¶

Frechet-type behavior is a statement about extremes, not necessarily about the full parent distribution. If $X_1,\dots,X_n$ are iid and the right tail is regularly varying with exponent $\alpha>0$, then an appropriate positive normalization $a_n$ puts the maximum

on the Frechet scale. The full regular-variation domain-of-attraction theorem is cited in the references; the exact Pareto calculation below shows the core mechanism.

For Pareto Type I with lower cutoff $x_m$ and exponent $\alpha$, take $a_n=x_m n^{1/\alpha}$. Then for fixed $y>0$ and all large enough $n$,

The same shape parameter is therefore seen in two coordinates:

Tail behavior¶

Although the Frechet law is a limit law for maxima, it is itself heavy-tailed. Its survival function satisfies

Thus the Frechet survival tail is regularly varying with index $-\alpha$. This is why the $\xi>0$ GEV regime is also the Pareto-type regime: the block-maximum limit and the parent survival tail share the same reciprocal coordinate $\xi=1/\alpha$.

Finite-block calculation¶

For any $\alpha>0$, the exact Pareto probability at $y=1$ is $(1-1/n)^n$. The limit is $e^{-1}$, independently of $\alpha$ because the normalization already includes $n^{1/\alpha}$.

For fixed $y>0$, set $t=y^{-\alpha}$. Expanding the logarithm gives

This explains the finite-block correction for an exact Pareto parent at a fixed $y$. It is not a uniform error bound over all thresholds or a rate for arbitrary regularly varying parents. No simulated or fitted CDF is used here.

Use frechet and the existing pareto_normalized_maximum_cdf.

from xmath.incerto.distributions import frechet
from xmath.incerto.glossary import pareto_normalized_maximum_cdf

for n in (1, 4, 16, 64):
    finite = pareto_normalized_maximum_cdf([1], alpha=1.5, n=n)[0]
    print(f"n={n:d}: cdf(1)={finite:.4g}, limit={frechet(1.5).cdf(1):.4g}")

Output

n=1: cdf(1)=0, limit=0.3679
n=4: cdf(1)=0.3164, limit=0.3679
n=16: cdf(1)=0.3561, limit=0.3679
n=64: cdf(1)=0.365, limit=0.3679

Regenerate both figures with uv run python demos/incerto/distribution_cards.py.

Caveats¶

• Frechet-type is a domain-of-attraction label. It does not say the original observations themselves follow a Frechet distribution.

• The $\xi>0$ conclusion is asymptotic. Finite samples can look Frechet-like over one range and deviate elsewhere because of body contamination, dependence, truncation, censoring, or second-order tail behavior.

• GEV block-maxima fitting uses a location-scale family. The unit Frechet law above is a convenient standard representative of the positive-shape class.

References¶

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

• Resnick, Heavy-Tail Phenomena 2007.

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

Backlinks¶

• Depends on: Generalized Extreme-Value Distribution, Pareto Distribution, and Regular Variation.

• Used by: Extreme Value Index Estimation and the positive-shape regime of Generalized Extreme-Value Distribution.