Generalized Pareto distribution

Generalized Pareto distribution
Generalized Pareto distribution
	Probability density function GPD distribution functions for and different values of and
	Cumulative distribution function
Parameters	location (real); scale (real); shape (real)
Support	;
PDF	; where
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF
CF
Method of moments	;

In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location $\mu$ , scale $\sigma$ , and shape $\xi$ . Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as $\kappa =-\xi \,$ .

Generalized Pareto distribution
Probability density function GPD distribution functions for $\mu =0$ and different values of $\sigma$ and $\xi$
Cumulative distribution function
Parameters	$\mu \in (-\infty ,\infty )\,$ location (real) $\sigma \in (0,\infty )\,$ scale (real) $\xi \in (-\infty ,\infty )\,$ shape (real)
Support	$x\geq \mu \,\;(\xi \geq 0)$ $\mu \leq x\leq \mu -\sigma /\xi \,\;(\xi <0)$
PDF	${\frac {1}{\sigma }}(1+\xi z)^{-(1/\xi +1)}$ where $z={\frac {x-\mu }{\sigma }}$
CDF	$1-(1+\xi z)^{-1/\xi }\,$
Mean	$\mu +{\frac {\sigma }{1-\xi }}\,\;(\xi <1)$
Median	$\mu +{\frac {\sigma (2^{\xi }-1)}{\xi }}$
Mode	$\mu$
Variance	${\frac {\sigma ^{2}}{\left(1-\xi \right)^{2}(1-2\xi )}}\,\;(\xi <1/2)$
Skewness	${\frac {2(1+\xi ){\sqrt {1-2\xi }}}{(1-3\xi )}}\,\;(\xi <1/3)$
Excess kurtosis	${\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)$
Entropy	$\log(\sigma )+\xi +1$
MGF	$e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$
CF	$e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$
Method of moments	$\xi ={\frac {1}{2}}\left(1-{\frac {\left(\operatorname {E} [X]-\mu \right)^{2}}{\operatorname {Var} [X]}}\right)$ $\sigma =(\operatorname {E} [X]-\mu )(1-\xi )$