Family of probability distributions often used to model tails or extreme values
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) |
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Support |

 |
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PDF |

where  |
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CDF |
 |
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Mean |
 |
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Median |
 |
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Mode |
 |
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Variance |
 |
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Skewness |
 |
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Ex. kurtosis |
 |
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Entropy |
 |
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MGF |
![{\displaystyle e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j)){\prod _{k=0}^{j}(1-k\xi )))\right],\;(k\xi <1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf9f358ac58dcba4130cba492879256576e783) |
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CF |
![{\displaystyle e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j)){\prod _{k=0}^{j}(1-k\xi )))\right],\;(k\xi <1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53bfef161abce3834ebc5908620389e3174d612f) |
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Method of Moments |
![{\displaystyle \sigma =(E[X]-\mu )(1-\xi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae5aff7c32202ca44e85df4abac26bc3e6deb14) |
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CVaR (ES) |
[1] |
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bPOE |
[1] |
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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
, scale
, and shape
.[2][3] Sometimes it is specified by only scale and shape[4] and sometimes only by its shape parameter. Some references give the shape parameter as
.[5]
Characterization
The related location-scale family of distributions is obtained by replacing the argument z by
and adjusting the support accordingly.
The cumulative distribution function of
(
,
, and
) is

where the support of
is
when
, and
when
.
The probability density function (pdf) of
is
,
again, for
when
, and
when
.
The pdf is a solution of the following differential equation:[citation needed]

Generating generalized Pareto random variables
Generating GPD random variables
If U is uniformly distributed on
(0, 1], then

and

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.

and

then

Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:
must be positive.
Exponentiated generalized Pareto distribution
The exponentiated generalized Pareto distribution (exGPD)
The pdf of the

(exponentiated generalized Pareto distribution) for different values

and

.
If

,
,
, then
is distributed according to the exponentiated generalized Pareto distribution, denoted by

,
.
The probability density function(pdf) of

,
is

where the support is
for
, and
for
.
For all
, the
becomes the location parameter. See the right panel for the pdf when the shape
is positive.
The exGPD has finite moments of all orders for all
and
.
The
variance of the

as a function of

. Note that the variance only depends on

. The red dotted line represents the variance evaluated at

, that is,

.
The moment-generating function of
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28884f7453a08deb806e6dcfadd72715427ba40b)
where
and
denote the beta function and gamma function, respectively.
The expected value of

,
depends on the scale
and shape
parameters, while the
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8417be06df13f42af281e304598ef2e687d03b5)
Note that for a fixed value for the
, the
plays as the location parameter under the exponentiated generalized Pareto distribution.
The variance of

,
depends on the shape parameter
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d91dba75f48dfd9845bc57efdc32455bacac8a)
See the right panel for the variance as a function of
. Note that
.
Note that the roles of the scale parameter
and the shape parameter
under
are separably interpretable, which may lead to a robust efficient estimation for the
than using the
[2]. The roles of the two parameters are associated each other under
(at least up to the second central moment); see the formula of variance
wherein both parameters are participated.
The Hill's estimator
Assume that
are
observations (not need to be i.i.d.) from an unknown heavy-tailed distribution
such that its tail distribution is regularly varying with the tail-index
(hence, the corresponding shape parameter is
). To be specific, the tail distribution is described as

It is of a particular interest in the extreme value theory to estimate the shape parameter
, especially when
is positive (so called the heavy-tailed distribution).
Let
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
, and large
,
is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate
: 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
, write
for the
-th largest value of
. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [3]) based on the
upper order statistics is defined as

In practice, the Hill estimator is used as follows. First, calculate the estimator
at each integer
, and then plot the ordered pairs
. Then, select from the set of Hill estimators
which are roughly constant with respect to
: these stable values are regarded as reasonable estimates for the shape parameter
. If
are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter
[4].
Note that the Hill estimator
makes a use of the log-transformation for the observations
. (The Pickand's estimator
also employed the log-transformation, but in a slightly different way
[5].)