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 fX(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
Support
Mean
Variance
| Example | α | β |
|---|---|---|
| Let X₁, ..., Xn 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)
E(X) = , Var(X) =
Note that this 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.
The illustration above shows a sample of n points chosen independently and at random from a Pareto(α, β) distribution. The distribution of the scaled maximum max Xi/n1/α converges to a Fréchet(α, β) distribution as n approaches infinity. Note that the maximum needs to be scaled 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 n1/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.