Extreme Value Distribution > Fréchet Distribution
What is the Fréchet Distribution?
The Fréchet Distribution, also called the extreme value distribution(EVD) Type II, is used to model maximum values in a data set. It is one of four EVDs in common use. The other three are the Gumbel Distribution, the Weibull Distribution and the Generalized Extreme Value Distribution. This distribution is used to model a wide range of phenomena like flood analysis, horse racing, human lifespans, maximum rainfalls and river discharges in hydrology.
The CDF for the Fréchet distribution is:
Pr(X≤x) = e-x-α.
The Fréchet distribution has a long, power-law tail* that slowly converges to 1. It has three parameters:
- Shape parameter, α,
- Scale parameter, β,
- Location parameter, μ.
Some texts and programs use slightly different notation, although the underlying principles are the same. For example, in some texts, the shape, location and scale parameters are denoted as ξ, m, and α respectively. In R, the shape, location and scale parameters are coded as s, α and β.
The general CDF (which included the location and scale parameters) is:
Pr(X ≤ x) = e-((x – μ)/β)α if x>μ.
The shape and scale parameters can be any real positive number and the location parameter can be any real number. This means that the distribution is defined for the interval (μ, ∞).
History
The Fréchet Distribution is named after French mathematician Maurice René Fréchet, who developed it in the 1920s as a maximum value distribution. It is equal to the reciprocal of the Weibull distribution, but they were developed separately. Fréchet went on to describe a Weibull-like distribution in 1927; This was used by Rosin and Rammler in 1933 who applied it to fit a particle size distribution. In 1951, Weibull took Fréchet’s inverse distribution, wrote an in-depth paper on it, and it took on his name.
Note: *According to Sornette(2003), any PDF with a tail falling as a power law X-1-μ for X → +∞ will have it’s extreme value distribution converging to the Fréchet with α=1/μ.
References:
- Fréchet, Maurice (1927), “Sur la loi de probabilité de l’écart maximum”, Annales de la Société Polonaise de Mathematique, Cracovie 6: 93–116.
- Rosin, P.; Rammler, E. (1933), “The Laws Governing the Fineness of Powdered Coal”, Journal of the Institute of Fuel 7: 29–36.
- Sornette, D. 2003. Critical Phenomena in Natural Sciences: Chaos, Fractals, Self organization.
- Weibull, W. (1951), “A statistical distribution function of wide applicability” (PDF),J. Appl. Mech.-Trans. ASME 18 (3): 293–297.