The standard cumulative distribution function (cdf) of the GPD is defined by^[6]

${\displaystyle F_{\xi }(z)={\begin{cases}1-\left(1+\xi z\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-e^{-z}&{\text{for }}\xi =0.\end{cases}}}$ 

where the support is ${\displaystyle z\geq 0}$  for ${\displaystyle \xi \geq 0}$  and ${\displaystyle 0\leq z\leq -1/\xi }$  for ${\displaystyle \xi <0}$ . The corresponding probability density function (pdf) is

${\displaystyle f_{\xi }(z)={\begin{cases}(1+\xi z)^{-{\frac {\xi +1}{\xi }}}&{\text{for }}\xi \neq 0,\\e^{-z}&{\text{for }}\xi =0.\end{cases}}}$ 

The related location-scale family of distributions is obtained by replacing the argument z by ${\displaystyle {\frac {x-\mu }{\sigma }}}$  and adjusting the support accordingly.

The cumulative distribution function of ${\displaystyle X\sim GPD(\mu ,\sigma ,\xi )}$  (${\displaystyle \mu \in \mathbb {R} }$ , ${\displaystyle \sigma >0}$ , and ${\displaystyle \xi \in \mathbb {R} }$ ) is

${\displaystyle F_{(\mu ,\sigma ,\xi )}(x)={\begin{cases}1-\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0,\end{cases}}}$ 

where the support of ${\displaystyle X}$  is ${\displaystyle x\geqslant \mu }$  when ${\displaystyle \xi \geqslant 0\,}$ , and ${\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi }$  when ${\displaystyle \xi <0}$ .

The probability density function (pdf) of ${\displaystyle X\sim GPD(\mu ,\sigma ,\xi )}$  is

${\displaystyle f_{(\mu ,\sigma ,\xi )}(x)={\frac {1}{\sigma }}\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{\left(-{\frac {1}{\xi }}-1\right)}}$ ,

again, for ${\displaystyle x\geqslant \mu }$  when ${\displaystyle \xi \geqslant 0}$ , and ${\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi }$  when ${\displaystyle \xi <0}$ .

The pdf is a solution of the following differential equation: ^{[citation needed]}

${\displaystyle \left\{{\begin{array}{l}f'(x)(-\mu \xi +\sigma +\xi x)+(\xi +1)f(x)=0,\\f(0)={\frac {\left(1-{\frac {\mu \xi }{\sigma }}\right)^{-{\frac {1}{\xi }}-1}}{\sigma }}\end{array}}\right\}}$ 

• If the shape ${\displaystyle \xi }$  and location ${\displaystyle \mu }$  are both zero, the GPD is equivalent to the exponential distribution.
• With shape ${\displaystyle \xi =-1}$ , the GPD is equivalent to the continuous uniform distribution ${\displaystyle U(0,\sigma )}$ .^[7]
• With shape ${\displaystyle \xi >0}$  and location ${\displaystyle \mu =\sigma /\xi }$ , the GPD is equivalent to the Pareto distribution with scale ${\displaystyle x_{m}=\sigma /\xi }$  and shape ${\displaystyle \alpha =1/\xi }$ .
• If ${\displaystyle X}$  ${\displaystyle \sim }$  ${\displaystyle GPD}$  ${\displaystyle (}$ ${\displaystyle \mu =0}$ , ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )}$ , then ${\displaystyle Y=\log(X)\sim exGPD(\sigma ,\xi )}$  [1]. (exGPD stands for the exponentiated generalized Pareto distribution.)
• GPD is similar to the Burr distribution.

Generating GPD random variables edit

If U is uniformly distributed on (0, 1], then

${\displaystyle X=\mu +{\frac {\sigma (U^{-\xi }-1)}{\xi }}\sim GPD(\mu ,\sigma ,\xi \neq 0)}$ 

and

${\displaystyle X=\mu -\sigma \ln(U)\sim GPD(\mu ,\sigma ,\xi =0).}$ 

Both formulas are obtained by inversion of the cdf.

In Matlab Statistics Toolbox, you can easily use "gprnd" command to generate generalized Pareto random numbers.

GPD as an Exponential-Gamma Mixture edit

A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter.

${\displaystyle X|\Lambda \sim \operatorname {Exp} (\Lambda )}$ 

and

${\displaystyle \Lambda \sim \operatorname {Gamma} (\alpha ,\beta )}$ 

then

${\displaystyle X\sim \operatorname {GPD} (\xi =1/\alpha ,\ \sigma =\beta /\alpha )}$ 

Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:${\displaystyle \xi }$  must be positive.

In addition to this mixture (or compound) expression, the generalized Pareto distribution can also be expressed as a simple ratio. Concretely, for ${\displaystyle Y\sim {\text{Exponential}}(1)}$  and ${\displaystyle Z\sim {\text{Gamma}}(1/\xi ,1)}$ , we have ${\displaystyle \mu +\sigma {\frac {Y}{\xi Z}}\sim {\text{GPD}}(\mu ,\sigma ,\xi )}$ . This is a consequence of the mixture after setting ${\displaystyle \beta =\alpha }$  and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.

The exponentiated generalized Pareto distribution (exGPD) edit

If ${\displaystyle X\sim GPD}$  ${\displaystyle (}$ ${\displaystyle \mu =0}$ , ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )}$ , then ${\displaystyle Y=\log(X)}$  is distributed according to the exponentiated generalized Pareto distribution, denoted by ${\displaystyle Y}$  ${\displaystyle \sim }$  ${\displaystyle exGPD}$  ${\displaystyle (}$ ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )}$ .

The probability density function(pdf) of ${\displaystyle Y}$  ${\displaystyle \sim }$  ${\displaystyle exGPD}$  ${\displaystyle (}$ ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )\,\,(\sigma >0)}$  is

${\displaystyle g_{(\sigma ,\xi )}(y)={\begin{cases}{\frac {e^{y}}{\sigma }}{\bigg (}1+{\frac {\xi e^{y}}{\sigma }}{\bigg )}^{-1/\xi -1}\,\,\,\,{\text{for }}\xi \neq 0,\\{\frac {1}{\sigma }}e^{y-e^{y}/\sigma }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi =0,\end{cases}}}$ 

where the support is ${\displaystyle -\infty <y<\infty }$  for ${\displaystyle \xi \geq 0}$ , and ${\displaystyle -\infty <y\leq \log(-\sigma /\xi )}$  for ${\displaystyle \xi <0}$ .

For all ${\displaystyle \xi }$ , the ${\displaystyle \log \sigma }$  becomes the location parameter. See the right panel for the pdf when the shape ${\displaystyle \xi }$  is positive.

The exGPD has finite moments of all orders for all ${\displaystyle \sigma >0}$  and ${\displaystyle -\infty <\xi <\infty }$ .

The moment-generating function of ${\displaystyle Y\sim exGPD(\sigma ,\xi )}$  is

${\displaystyle M_{Y}(s)=E[e^{sY}]={\begin{cases}-{\frac {1}{\xi }}{\bigg (}-{\frac {\sigma }{\xi }}{\bigg )}^{s}B(s+1,-1/\xi )\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}s\in (-1,\infty ),\xi <0,\\{\frac {1}{\xi }}{\bigg (}{\frac {\sigma }{\xi }}{\bigg )}^{s}B(s+1,1/\xi -s)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}s\in (-1,1/\xi ),\xi >0,\\\sigma ^{s}\Gamma (1+s)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}s\in (-1,\infty ),\xi =0,\end{cases}}}$ 

where ${\displaystyle B(a,b)}$  and ${\displaystyle \Gamma (a)}$  denote the beta function and gamma function, respectively.

The expected value of ${\displaystyle Y}$  ${\displaystyle \sim }$  ${\displaystyle exGPD}$  ${\displaystyle (}$ ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )}$  depends on the scale ${\displaystyle \sigma }$  and shape ${\displaystyle \xi }$  parameters, while the ${\displaystyle \xi }$  participates through the digamma function:

${\displaystyle E[Y]={\begin{cases}\log \ {\bigg (}-{\frac {\sigma }{\xi }}{\bigg )}+\psi (1)-\psi (-1/\xi +1)\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi <0,\\\log \ {\bigg (}{\frac {\sigma }{\xi }}{\bigg )}+\psi (1)-\psi (1/\xi )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi >0,\\\log \sigma +\psi (1)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi =0.\end{cases}}}$ 

Note that for a fixed value for the ${\displaystyle \xi \in (-\infty ,\infty )}$ , the ${\displaystyle \log \ \sigma }$  plays as the location parameter under the exponentiated generalized Pareto distribution.

The variance of ${\displaystyle Y}$  ${\displaystyle \sim }$  ${\displaystyle exGPD}$  ${\displaystyle (}$ ${\displaystyle \sigma }$ , ${\displaystyle \xi }$  ${\displaystyle )}$  depends on the shape parameter ${\displaystyle \xi }$  only through the polygamma function of order 1 (also called the trigamma function):

${\displaystyle Var[Y]={\begin{cases}\psi '(1)-\psi '(-1/\xi +1)\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi <0,\\\psi '(1)+\psi '(1/\xi )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi >0,\\\psi '(1)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for }}\xi =0.\end{cases}}}$ 

See the right panel for the variance as a function of ${\displaystyle \xi }$ . Note that ${\displaystyle \psi '(1)=\pi ^{2}/6\approx 1.644934}$ .

Note that the roles of the scale parameter ${\displaystyle \sigma }$  and the shape parameter ${\displaystyle \xi }$  under ${\displaystyle Y\sim exGPD(\sigma ,\xi )}$  are separably interpretable, which may lead to a robust efficient estimation for the ${\displaystyle \xi }$  than using the ${\displaystyle X\sim GPD(\sigma ,\xi )}$  [2]. The roles of the two parameters are associated each other under ${\displaystyle X\sim GPD(\mu =0,\sigma ,\xi )}$  (at least up to the second central moment); see the formula of variance ${\displaystyle Var(X)}$  wherein both parameters are participated.

Assume that ${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}$  are ${\displaystyle n}$  observations (not need to be i.i.d.) from an unknown heavy-tailed distribution ${\displaystyle F}$  such that its tail distribution is regularly varying with the tail-index ${\displaystyle 1/\xi }$  (hence, the corresponding shape parameter is ${\displaystyle \xi }$ ). To be specific, the tail distribution is described as

${\displaystyle {\bar {F}}(x)=1-F(x)=L(x)\cdot x^{-1/\xi },\,\,\,\,\,{\text{for some }}\xi >0,\,\,{\text{where }}L{\text{ is a slowly varying function.}}}$ 

It is of a particular interest in the extreme value theory to estimate the shape parameter ${\displaystyle \xi }$ , especially when ${\displaystyle \xi }$  is positive (so called the heavy-tailed distribution).

Let ${\displaystyle F_{u}}$  be their conditional excess distribution function. Pickands–Balkema–de Haan theorem (Pickands, 1975; Balkema and de Haan, 1974) states that for a large class of underlying distribution functions ${\displaystyle F}$ , and large ${\displaystyle u}$ , ${\displaystyle F_{u}}$  is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate ${\displaystyle \xi }$ : the GPD plays the key role in POT approach.

A renowned estimator using the POT methodology is the Hill's estimator. Technical formulation of the Hill's estimator is as follows. For ${\displaystyle 1\leq i\leq n}$ , write ${\displaystyle X_{(i)}}$  for the ${\displaystyle i}$ -th largest value of ${\displaystyle X_{1},\cdots ,X_{n}}$ . Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [3]) based on the ${\displaystyle k}$  upper order statistics is defined as

${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}={\widehat {\xi }}_{k}^{\text{Hill}}(X_{1:n})={\frac {1}{k-1}}\sum _{j=1}^{k-1}\log {\bigg (}{\frac {X_{(j)}}{X_{(k)}}}{\bigg )},\,\,\,\,\,\,\,\,{\text{for }}2\leq k\leq n.}$ 

In practice, the Hill estimator is used as follows. First, calculate the estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}$  at each integer ${\displaystyle k\in \{2,\cdots ,n\}}$ , and then plot the ordered pairs ${\displaystyle \{(k,{\widehat {\xi }}_{k}^{\text{Hill}})\}_{k=2}^{n}}$ . Then, select from the set of Hill estimators ${\displaystyle \{{\widehat {\xi }}_{k}^{\text{Hill}}\}_{k=2}^{n}}$  which are roughly constant with respect to ${\displaystyle k}$ : these stable values are regarded as reasonable estimates for the shape parameter ${\displaystyle \xi }$ . If ${\displaystyle X_{1},\cdots ,X_{n}}$  are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter ${\displaystyle \xi }$  [4].

Note that the Hill estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}$  makes a use of the log-transformation for the observations ${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}$ . (The Pickand's estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Pickand}}}$  also employed the log-transformation, but in a slightly different way [5].)

• Burr distribution
• Pareto distribution
• Generalized extreme value distribution
• Exponentiated generalized Pareto distribution
• Pickands–Balkema–de Haan theorem

• Pickands, James (1975). "Statistical inference using extreme order statistics" (PDF). Annals of Statistics. 3 s: 119–131. doi:10.1214/aos/1176343003.
• Balkema, A.; De Haan, Laurens (1974). "Residual life time at great age". Annals of Probability. 2 (5): 792–804. doi:10.1214/aop/1176996548.
• Lee, Seyoon; Kim, J.H.K. (2018). "Exponentiated generalized Pareto distribution:Properties and applications towards extreme value theory". Communications in Statistics - Theory and Methods. 48 (8): 1–25. arXiv:1708.01686. doi:10.1080/03610926.2018.1441418. S2CID 88514574.
• N. L. Johnson; S. Kotz; N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, second edition. New York: Wiley. ISBN 978-0-471-58495-7. Chapter 20, Section 12: Generalized Pareto Distributions.
• Barry C. Arnold (2011). "Chapter 7: Pareto and Generalized Pareto Distributions". In Duangkamon Chotikapanich (ed.). Modeling Distributions and Lorenz Curves. New York: Springer. ISBN 9780387727967.
• Arnold, B. C.; Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics.

• Mathworks: Generalized Pareto distribution