# Fréchet distribution

Parameters Probability density function Cumulative distribution function $\alpha \in (0,\infty)$ shape. (Optionally, two more parameters) $s \in (0,\infty)$ scale (default: $s=1 \,$) $m \in (-\infty,\infty)$ location of minimum (default: $m=0 \,$) $x>m$ $\frac{\alpha}{s} \; \left(\frac{x-m}{s}\right)^{-1-\alpha} \; e^{-(\frac{x-m}{s})^{-\alpha}}$ $e^{-(\frac{x-m}{s})^{-\alpha}}$ $\begin{cases} \ m+s\Gamma\left(1-\frac{1}{\alpha}\right) & \text{for } \alpha>1 \\ \ \infty & \text{otherwise} \end{cases}$ $m+\frac{s}{\sqrt[\alpha]{\log_e(2)}}$ $m+s\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}$ $\begin{cases} \ s^2\left(\Gamma\left(1-\frac{2}{\alpha}\right)- \left(\Gamma\left(1-\frac{1}{\alpha}\right)\right)^2\right) & \text{for } \alpha>2 \\ \ \infty & \text{otherwise} \end{cases}$ $\begin{cases} \ \frac{\Gamma\left(1-\frac {3}{\alpha}\right)-3\Gamma\left(1-\frac {2}{\alpha}\right)\Gamma\left(1-\frac {1}{\alpha}\right)+2\Gamma^3\left(1-\frac {1}{\alpha} \right)}{\sqrt{ \left( \Gamma\left(1-\frac{2}{\alpha}\right)-\Gamma^2\left(1-\frac{1}{\alpha}\right) \right)^3 }} & \text{for } \alpha>3 \\ \ \infty & \text{otherwise} \end{cases}$ $\begin{cases} \ -6+ \frac{\Gamma \left(1-\frac{4}{\alpha}\right) -4\Gamma\left(1-\frac{3}{\alpha}\right) \Gamma\left(1-\frac{1}{\alpha}\right)+3 \Gamma^2\left(1-\frac{2}{\alpha} \right)} {\left[\Gamma \left(1-\frac{2}{\alpha}\right) - \Gamma^2 \left(1-\frac{1}{\alpha}\right) \right]^2} & \text{for } \alpha>4 \\ \ \infty & \text{otherwise} \end{cases}$ $1 + \frac{\gamma}{\alpha} + \gamma +\ln \left( \frac{s}{\alpha} \right)$, where $\gamma$ is the Euler–Mascheroni constant. [1] Note: Moment $k$ exists if $\alpha>k$ [1]

The Fréchet distribution is a special case of the generalized extreme value distribution. It has the cumulative distribution function

$\Pr(X \le x)=e^{-x^{-\alpha}} \text{ if } x>0.$

where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function

$\Pr(X \le x)=e^{-\left(\frac{x-m}{s}\right)^{-\alpha}} \text{ if } x>m.$

Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.

## Characteristics

The single parameter Fréchet with parameter $\alpha$ has standardized moment

$\mu_k=\int_0^\infty x^k f(x)dx=\int_0^\infty t^{-\frac{k}{\alpha}}e^{-t} \, dt,$

(with $t=x^{-\alpha}$) defined only for $k<\alpha$:

$\mu_k=\Gamma\left(1-\frac{k}{\alpha}\right)$

where $\Gamma\left(z\right)$ is the Gamma function.

In particular:

• For $\alpha>1$ the expectation is $E[X]=\Gamma(1-\tfrac{1}{\alpha})$
• For $\alpha>2$ the variance is $\text{Var}(X)=\Gamma(1-\tfrac{2}{\alpha})-\big(\Gamma(1-\tfrac{1}{\alpha})\big)^2.$

The quantile $q_y$ of order $y$ can be expressed through the inverse of the distribution,

$q_y=F^{-1}(y)=\left(-\log_e y \right)^{-\frac{1}{\alpha}}$.

In particular the median is:

$q_{1/2}=(\log_e 2)^{-\frac{1}{\alpha}}.$

The mode of the distribution is $\left(\frac{\alpha}{\alpha+1}\right)^\frac{1}{\alpha}.$

Especially for the 3-parameter Fréchet, the first quartile is $q_1= m+\frac{s}{\sqrt[\alpha]{\log(4)}}$ and the third quartile $q_3= m+\frac{s}{\sqrt[\alpha]{\log(\frac{4}{3})}}.$

Also the quantiles for the mean and mode are:

$F(mean)=\exp \left( -\Gamma^{-\alpha} \left(1- \frac{1}{\alpha} \right) \right)$
$F(mode)=\exp \left( -\frac{\alpha+1}{\alpha} \right).$
Fitted cumulative Fréchet distribution to extreme one-day rainfalls

## Related distributions

• If $X \sim U(0,1) \,$ (Uniform distribution (continuous)) then $m + s(-\log(X))^{-1/\alpha} \sim \textrm{Frechet}(\alpha,s,m)\,$
• If $X \sim \textrm{Frechet}(\alpha,s,m)\,$ then $k X + b \sim \textrm{Frechet}(\alpha,k s,k m + b)\,$
• If $X_i=\textrm{Frechet}(\alpha,s,m) \,$ and $Y=\max\{\,X_1,\ldots,X_n\,\} \,$ then $Y \sim \textrm{Frechet}(\alpha,n^{\tfrac{1}{\alpha}} s,m) \,$
• The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
• If $X \sim \textrm{Weibull}(k=\alpha, \lambda=m)\,$ (Weibull distribution) then $\tfrac{m^2}{X} \sim \textrm{Frechet}(\alpha,m)\,$