Generalized Pareto distribution

Generalized Pareto distribution is defined by three parameters denoted $k$ , $\sigma$  and $\theta$ . The probability density function depends from the value of the parameters, several cases have to be distinguished.

• $k > 0$

The support of the distribution is $[\theta, \infty]$. The probability density function is

$f(x) = \left(\frac{1}{\sigma}\right) \left(1 + k \frac{x- \theta}{\sigma} \right) ^{-1- \frac{1}{k}}$

• $k < 0$

The support of the distribution is $[ -\infty, \theta]$. The probability density function is

$f(x) = \left(\frac{1}{\sigma}\right) \left(1 + k \frac{x- \theta}{\sigma} \right) ^{-1- \frac{1}{k}}$
• $k = 0$ ,

The support of the distribution is $[ -\infty, \infty]$. The probability density function is:

$f(x) = \left(\frac{1}{\sigma}\right) exp \left( \frac{x- \theta}{\sigma} \right)$