Extreme Value Distributions (Haskell)

The family of Extreme Value Distributions in Haskell.

Basic Distributional Quantities (CDF, PDF, Quantile and Random Generation) for the Gumbel, Fréchet, (inverse) Weibull and GEV Distributions.

Installation • Details

Installation

You can install the package with cabal install gev-lib or adding the package directly in the .cabal file of your project.

Details

We quickly present the distributions in question. You can read more about the family of GEV Distributions here.

Gumbel Distribution

The Gumbel distribution is defined, for location parameter $\mu \in \mathbb{R}$ and scale parameter $\sigma > 0$, with the CDF $$F(x) = \exp \left ( - \exp \left ( - \frac{x - \mu}{\sigma} \right ) \right ), \quad x \in \mathbb{R}. $$

Fréchet Distribution

The Fréchet distribution is defined, for location parameter $\mu \in \mathbb{R}$, scale and shape parameters $\sigma, \zeta >0$, with the CDF $$F(x) = \exp \left ( - \left ( \frac{x - \mu}{\sigma} \right)^{-\zeta} \right ), \quad x \in \mathbb{R}. $$

(Inverse) Weibull Distribution

The Weibull Distribution, which is in fact the Inverse Weibull distribution, is defined for location parameter $\mu \in \mathbb{R}$, scale and shape parameters $\sigma, \zeta >0$, with the CDF $$F(x) = \exp \left ( - \left ( - \left ( \frac{x - \mu}{ \sigma } \right) \right)^{\zeta} \right ), \quad x \in \mathbb{R}. $$

GEV Distribution

The GEV Distribution generalizes all of the above distributions. It is defined for location parameter $\mu \in \mathbb{R}$, scale parameter $\sigma >0$ and shape parameters $\zeta \in \mathbb{R}$ with the CDF of $F(x) = \exp \left ( t(x) \right)$ for $1 + \zeta \left( \frac{x - \mu}{\sigma} \right ) > 0$, where $$t(x) = \left( 1 + \zeta \left( \frac{x - \mu}{\sigma} \right) \right)^{- \frac{1}{\zeta}} \quad \text{if} \quad \zeta \neq 0, $$ and $$t(x) = \exp \left ( - \frac{x - \mu}{\sigma} \right ) \quad \text{if} \quad \zeta = 0.$$

In fact, for $\zeta = 0$ we recover the Gumbel distribution, for $\zeta > 0$ we recover the Fréchet distribution and for $\zeta < 0$ we have the Weibull distribution.

To do

• Write better documentation.
• Add Examples.