< List of probability distributions
Statistics is an integral part of any scientific research, and its applications are diverse. One of the important distributions in statistics is the Fréchet distribution, which is also known as the Extreme Value Distribution (EVD) Type II or the inverse Weibull distribution. It is used to model maximum values in a dataset, which is useful in many fields, including hydrology, finance, and sports. In this article, we will explore the Fréchet distribution, including its properties, applications, and how it differs from other EVDs.
About the Fréchet distribution
The Fréchet is one of three types of EVDs with different properties:
- Type I EVD (Gumbel distribution) is skewed towards one side with long tails.
- Type II EVD (Fréchet Distribution) is symmetric close to its mean value (μ).
- Type III EVD (Weibull distribution) has long tails on both sides close to the mean but has smaller variations near its middle.
The Fréchet distribution is named after Maurice Fréchet, a French mathematician who introduced it in 1927 [2]. Fréchet also described a Weibull-like distribution in 1927; Rosin and Rammler applied it to fit a particle size distribution in 1933 [3]. In 1951, Weibull took Fréchet’s “inverse” distribution, wrote an in-depth paper on it [4], and it took on his name.
The probability density function (pdf) of the Fréchet distribution is given by [5]
where the support is 0 ≤ x < ∞ and
- α, the shape parameter (α > 0);
- β, the scale parameter (β > 0).
The cumulative distribution function (CDF) for the Fréchet distribution is:
Pr(X ≤ x) = e-x-α, for x > 0
The mean is
and the variance is
Some texts note a three parameter distribution with an additional parameter μ for the location. The shape, location and scale parameters are also sometimes denoted as ξ, m, and α respectively. In R, the shape, location and scale parameters are coded as s, α and β.
The general three-parameter CDF is:
Pr(X ≤ x) = exp -((x – μ) / β)α if x > μ.
Fréchet distribution applications
The Fréchet distribution is used to model extreme events, such as floods, hurricanes, and earthquake magnitudes. It describes the probability of observing a maximum value, given a certain underlying distribution. The shape parameter α governs the tail behavior of the distribution. When α 1, the tail is light. When α = 1, the distribution reduces to the Weibull distribution (EVD Type III).
One of the applications of the Fréchet distribution is in flood analysis. Floods are catastrophic events that cause extensive damage to property and lives. By modeling the maximum flow of rivers, the risk of flooding can be quantified, and floodplains can be identified. The Fréchet is also used in finance to model extreme returns on investments, and in sports to predict the maximum performance of athletes.
Compared to other EVDs, the Fréchet has a heavier tail than the Gumbel and Weibull distributions but lighter than the generalized extreme value distribution (GEV). The Gumbel distribution is used to model the minimum values in a dataset, while the Weibull distribution is used to model the failure times of mechanical systems. The GEV is a more general distribution that includes the other three as special cases.
In conclusion, the Fréchet distribution is a valuable tool in statistics for modeling extreme events. Its applications are diverse and include hydrology, finance, and sports. By understanding the Fréchet distribution, researchers can better analyze and interpret data that includes maximum values.
References
[1] Image of EVD: R D Gill, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons
[2] Fréchet, Maurice (1927), “Sur la loi de probabilité de l’écart maximum”, Annales de la Société Polonaise de Mathematique, Cracovie 6: 93–116.
[3] Rosin, P.; Rammler, E. (1933), “The Laws Governing the Fineness of Powdered Coal”, Journal of the Institute of Fuel 7: 29–36.
[4] Cunningham, A. Probability playground: The Fréchet Distribution. Retrieved April 19, 2023 from: https://www.acsu.buffalo.edu/~adamcunn/probability/frechet.html
[5] Weibull, W. (1951), “A statistical distribution function of wide applicability” (PDF),J. Appl. Mech.-Trans. ASME 18 (3): 293–297.