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 $\mu$, 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 $\kappa = - \xi \,$.

Definition

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

: $F_(z) = \begin 1 - \left(1 + \xi z\right)^ & \text\xi \neq 0, \\ 1 - e^ & \text\xi = 0. \end$

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

: $f_(z) = \begin (1 + \xi z)^ & \text\xi \neq 0, \\ e^ & \text\xi = 0. \end$

Characterization

The related location-scale family of distributions is obtained by replacing the argument ''z'' by $\frac$ and adjusting the support accordingly.

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

: $F_(x) = \begin 1 - \left(1+ \frac\right)^ & \text\xi \neq 0, \\ 1 - \exp \left(-\frac\right) & \text\xi = 0, \end$
where the support of $X$ is $x \geqslant \mu$ when $\xi \geqslant 0 \,$, and $\mu \leqslant x \leqslant \mu - \sigma /\xi$ when $\xi < 0$.

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

: $f_(x) = \frac\left(1 + \frac\right)^$,

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

The pdf is a solution of the following differential equation:

:$\left\$

Special cases

*If the shape $\xi$ and location $\mu$ are both zero, the GPD is equivalent to the exponential distribution.
*With shape $\xi = -1$, the GPD is equivalent to the continuous uniform distribution $U(0, \sigma)$. Castillo, Enrique, and Ali S. Hadi. "Fitting the generalized Pareto distribution to data." Journal of the American Statistical Association 92.440 (1997): 1609-1620.
*With shape $\xi > 0$ and location $\mu = \sigma/\xi$, the GPD is equivalent to the Pareto distribution with scale $x_m=\sigma/\xi$ and shape $\alpha=1/\xi$.
*If $X$ $\sim$ $GPD$ $($$\mu = 0$, $\sigma$, $\xi$ $)$, then Y = \log (X) \sim exGPD(\sigma, \xi)
(exGPD stands for the Generalized Pareto distribution#Exponentiated generalized Pareto distribution, 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

:$X = \mu + \frac \sim GPD(\mu, \sigma, \xi \neq 0)$
and
:$X = \mu - \sigma \ln(U) \sim GPD(\mu,\sigma,\xi =0).$

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.

:$X, \Lambda \sim \operatorname(\Lambda)$
and
:$\Lambda \sim \operatorname(\alpha, \beta)$
then
:$X \sim \operatorname(\xi = 1/\alpha, \ \sigma = \beta/\alpha)$

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

Exponentiated generalized Pareto distribution

The exponentiated generalized Pareto distribution (exGPD)

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

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

:$g_(y) = \begin \frac\bigg( 1 + \frac \bigg)^\,\,\,\, \text \xi \neq 0, \\ \frace^ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\, \text \xi = 0 ,\end$
where the support is $-\infty < y < \infty$ for $\xi \geq 0$, and $-\infty < y \leq \log(-\sigma/\xi)$ for $\xi < 0$.

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

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

The moment-generating function of $Y \sim exGPD(\sigma,\xi)$ is
: M_Y(s) = E ^= \begin -\frac\bigg(-\frac\bigg)^ B(s+1, -1/\xi) \,\,\,\,\,\,\,\,\,\,\,\, \text s \in (-1, \infty), \xi < 0 , \\
\frac\bigg(\frac\bigg)^ B(s+1, 1/\xi - s) \,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, \text s \in (-1, 1/\xi), \xi > 0 , \\
\sigma^ \Gamma(1+s) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text s \in (-1, \infty), \xi = 0, \end
where $B(a,b)$ and $\Gamma (a)$ denote the beta function and gamma function, respectively.

The expected value of $Y$ $\sim$ $exGPD$ $($$\sigma$, $\xi$ $)$ depends on the scale $\sigma$ and shape $\xi$ parameters, while the $\xi$ participates through the digamma function:
: E = \begin \log\ \bigg(-\frac \bigg)+ \psi(1) - \psi(-1/\xi+1) \,\,\,\,\,\,\,\,\,\,\,\, \,\, \text\xi < 0 , \\
\log\ \bigg(\frac \bigg)+ \psi(1) - \psi(1/\xi) \,\,\,\,\,\,\,\,\,\,\,\, \,\,\, \,\,\, \,\,\, \,\,\, \,\,\, \,\,\,\,\,\, \,\,\, \text\xi > 0 , \\
\log \sigma + \psi(1) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\, \,\,\, \,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\, \,\,\,\,\,\,\, \text\xi = 0. \end
Note that for a fixed value for the $\xi \in (-\infty,\infty)$, the $\log\ \sigma$ plays as the location parameter under the exponentiated generalized Pareto distribution.

The variance of $Y$ $\sim$ $exGPD$ $($$\sigma$, $\xi$ $)$ depends on the shape parameter $\xi$ only through the polygamma function of order 1 (also called the trigamma function):
: Var = \begin \psi'(1) - \psi'(-1/\xi +1) \,\,\,\,\,\,\,\,\,\,\,\, \, \text\xi < 0 , \\
\psi'(1) + \psi'(1/\xi) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text\xi > 0 , \\
\psi'(1) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\text\xi = 0. \end
See the right panel for the variance as a function of $\xi$. Note that $\psi'(1) = \pi^2/6 \approx 1.644934$.

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

The Hill's estimator

Assume that $X_ = (X_1, \cdots, X_n)$ are $n$ observations (not need to be i.i.d.) from an unknown heavy-tailed distribution $F$ such that its tail distribution is regularly varying with the tail-index $1/\xi$ (hence, the corresponding shape parameter is $\xi$). To be specific, the tail distribution is described as
:$\bar(x) = 1 - F(x) = L(x) \cdot x^, \,\,\,\,\,\text\xi>0,\,\,\text L \text$
It is of a particular interest in the extreme value theory to estimate the shape parameter $\xi$, especially when $\xi$ is positive (so called the heavy-tailed distribution).

Let $F_u$ 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 $F$, and large $u$, $F_u$ is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate $\xi$: ''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 $1\leq i \leq n$, write $X_$ for the $i$-th largest value of $X_1, \cdots, X_n$. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et a

based on the $k$ upper order statistics is defined as
:$\widehat_^ = \widehat_^(X_) = \frac \sum_^ \log \bigg(\frac \bigg), \,\,\,\,\,\,\,\, \text 2 \leq k \leq n.$
In practice, the Hill estimator is used as follows. First, calculate the estimator $\widehat_^$ at each integer $k \in \$, and then plot the ordered pairs $\_^$. Then, select from the set of Hill estimators $\_^$ which are roughly constant with respect to $k$: these stable values are regarded as reasonable estimates for the shape parameter $\xi$. If $X_1, \cdots, X_n$ are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter \xi

Note that the Hill estimator $\widehat_^$ makes a use of the log-transformation for the observations $X_ = (X_1, \cdots, X_n)$. (The Pickand's estimator $\widehat_^$ also employed the log-transformation, but in a slightly different wa

)

See also

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

References

Further reading

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* Chapter 20, Section 12: Generalized Pareto Distributions.
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External links

Mathworks: Generalized Pareto distribution

{{ProbDistributions, continuous-variable

Continuous distributions
Power laws
Probability distributions with non-finite variance