Probability Playground: The Fréchet Distribution

The Fréchet distribution (also known as the inverse Weibull distribution) is used to model the distribution of the maximum value of a sample. It is therefore a type of extreme value distribution (type II), along with the Gumbel and Weibull.

It can be shown that the suitably normalized maximum of any sample {Xᵢ} will converge to a Fréchet random variable as n → ∞ if the pdf f_X(x) falls according to a power law as x → ∞.

Parameter	Range	Description
α	α > 0	Shape parameter
β	β > 0	Scale parameter

Probability Density Function

f (x; α, β) = \frac{α}{β} {(\frac{x}{β})}^{- α - 1} exp [- {(\frac{x}{β})}^{- α}]

F (x; α, β) = exp [- {(\frac{x}{β})}^{- α}]

S (x; α, β) = 1 - exp [- {(\frac{x}{β})}^{- α}]

h (x; α, β) = \frac{\frac{α}{β} {(\frac{x}{β})}^{- α - 1} exp [- {(\frac{x}{β})}^{- α}]}{1 - exp [- {(\frac{x}{β})}^{- α}]}

Support

0 < x < \infty

Mean

Variance

Example	α	β
Let X₁, ..., X_n be a sample drawn from a standard Pareto distribution. Then the distribution of max{Xᵢ/n} converges to a Fréchet(1, 1) distribution as n → ∞.	1.000	1.000
Recovery times (in days) from concussion in adults follow an approximate Fréchet(2.1, 8.37) distribution.	2.100	8.370
The largest annual earthquakes (on the Richter scale) for the Eastern United States for 1932 - 2021 follow an approximate Fréchet(9.87, 4.11) distribution.	9.870	4.110

X ∼ Fréchet(α, β)

α = β =

E(X) = , Var(X) =

Note that the pdf of the distribution is highly right-skewed, with a steep left side and a long right tail. The mean is only defined for α > 1, while the variance is only defined for α > 2.

Since the pdf of the distribution is highly right-skewed, the cdf rise steeply at first then much more slowly.

The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).

Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.

The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.

The illustration above shows a sample of n points chosen independently and at random from a Pareto(α, β) distribution. The distribution of max X_i/n^1/α converges to a Fréchet(α, β) distribution as n approaches infinity. Note that the maximum needs to be scaled by a factor of n^1/α since it otherwise increases without bound as n increases.

The simulation above shows a sample of n points (marked red) chosen independently and at random from a Pareto(α, β) distribution. The light blue line shows the maximum value of the sample divided by n^1/α. The distribution of this scaled maximum converges to a Fréchet(α, β) distribution as n approaches infinity. The histogram accumulates the results of each simulation.

Y = max Xᵢ ∼ Fréchet(α, βn^1/α) 1/X ∼ Weibull(α, 1/β)

E(Y) = , Var(Y) =

Proof

X ∼ Fréchet(α, β)

Y = max Xᵢ ∼ Fréchet(α, βn1/α) 1/X ∼ Weibull(α, 1/β)

Y = max Xᵢ ∼ Fréchet(α, βn^1/α) 1/X ∼ Weibull(α, 1/β)