Generalized Pareto distribution

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 ${\displaystyle \mu }$, scale ${\displaystyle \sigma }$, and shape ${\displaystyle \xi }$.[1][2] Sometimes it is specified by only scale and shape[3] and sometimes only by its shape parameter. Some references give the shape parameter as ${\displaystyle \kappa =-\xi \,}$.[4]

Parameters Probability density functionGPD distribution functions for ${\displaystyle \mu =0}$ and different values of ${\displaystyle \sigma }$ and ${\displaystyle \xi }$ Cumulative distribution function ${\displaystyle \mu \in (-\infty ,\infty )\,}$ location (real)${\displaystyle \sigma \in (0,\infty )\,}$ scale (real) ${\displaystyle \xi \in (-\infty ,\infty )\,}$ shape (real) ${\displaystyle x\geqslant \mu \,\;(\xi \geqslant 0)}$ ${\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi \,\;(\xi <0)}$ ${\displaystyle {\frac {1}{\sigma }}(1+\xi z)^{-(1/\xi +1)}}$ where ${\displaystyle z={\frac {x-\mu }{\sigma }}}$ ${\displaystyle 1-(1+\xi z)^{-1/\xi }\,}$ ${\displaystyle \mu +{\frac {\sigma }{1-\xi }}\,\;(\xi <1)}$ ${\displaystyle \mu +{\frac {\sigma (2^{\xi }-1)}{\xi }}}$ ${\displaystyle {\frac {\sigma ^{2}}{(1-\xi )^{2}(1-2\xi )}}\,\;(\xi <1/2)}$ ${\displaystyle {\frac {2(1+\xi ){\sqrt {1-2\xi }}}{(1-3\xi )}}\,\;(\xi <1/3)}$ ${\displaystyle {\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)}$ ${\displaystyle \log(\sigma )+\xi +1}$ ${\displaystyle e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)}$ ${\displaystyle e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)}$ ${\displaystyle \xi ={\frac {1}{2}}\left(1-{\frac {(E[X]-\mu )^{2}}{V[X]}}\right)}$ ${\displaystyle \sigma =(E[X]-\mu )(1-\xi )}$

Definition

The standard cumulative distribution function (cdf) of the GPD is defined by[5]

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

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

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

Characterization

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]

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

Special cases

• If the shape ${\displaystyle \xi }$  and location ${\displaystyle \mu }$  are both zero, the GPD is equivalent to the exponential distribution.
• 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 generalized Pareto random variables

Generating GPD random variables

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

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

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

and

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

then

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

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.

Exponentiated generalized Pareto distribution

The exponentiated generalized Pareto distribution (exGPD)

The pdf of the ${\displaystyle exGPD(\sigma ,\xi )}$  (exponentiated generalized Pareto distribution) for different values ${\displaystyle \sigma }$  and ${\displaystyle \xi }$ .

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   for ${\displaystyle \xi \geq 0}$ , and ${\displaystyle -\infty   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 variance of the ${\displaystyle exGPD(\sigma ,\xi )}$  as a function of ${\displaystyle \xi }$ . The red dotted line corresponds to the value of variance (${\displaystyle \psi ^{'}(1)=\pi ^{2}/6}$ ) evaluated at ${\displaystyle \xi =0}$ .

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 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.

The Hill's estimator

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].)

References

1. ^ Coles, Stuart (2001-12-12). An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75. ISBN 9781852334598.
2. ^ Dargahi-Noubary, G. R. (1989). "On tail estimation: An improved method". Mathematical Geology. 21 (8): 829–842. doi:10.1007/BF00894450. S2CID 122710961.
3. ^ Hosking, J. R. M.; Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distribution". Technometrics. 29 (3): 339–349. doi:10.2307/1269343. JSTOR 1269343.
4. ^ Davison, A. C. (1984-09-30). "Modelling Excesses over High Thresholds, with an Application". In de Oliveira, J. Tiago (ed.). Statistical Extremes and Applications. Kluwer. p. 462. ISBN 9789027718044.
5. ^ Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance. p. 162. ISBN 9783540609315.