Generalized Pareto distribution

The standard cumulative distribution function (cdf) of the GPD is defined by{{Cite book |last1=Embrechts |first1=Paul |last2=Klüppelberg |first2=Claudia|author2-link= Claudia Klüppelberg |last3=Mikosch |first3=Thomas |title=Modelling extremal events for insurance and finance |page=162 |url=https://books.google.com/books?id=BXOI2pICfJUC |isbn=9783540609315 |date=1997-01-01|publisher=Springer }}

: $F\_\{\backslash xi\}(z)\; =\; \backslash begin\{cases\}$

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 $z\; \backslash geq\; 0$ for $\backslash xi\; \backslash geq\; 0$ and $0\; \backslash leq\; z\; \backslash leq\; -\; 1\; /\backslash xi$ for . The corresponding probability density function (pdf) is

: $f\_\{\backslash xi\}(z)\; =\; \backslash begin\{cases\}$

(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 $\backslash frac\{x-\backslash mu\}\{\backslash sigma\}$ and adjusting the support accordingly.

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

: $F\_\{(\backslash mu,\backslash sigma,\backslash xi)\}(x)\; =\; \backslash begin\{cases\}$

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 $X$ is $x\; \backslash geqslant\; \backslash mu$ when $\backslash xi\; \backslash geqslant\; 0\; \backslash ,$, and $\backslash mu\; \backslash leqslant\; x\; \backslash leqslant\; \backslash mu\; -\; \backslash sigma\; /\backslash xi$ when .

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

: $f\_\{(\backslash mu,\backslash sigma,\backslash xi)\}(x)\; =\; \backslash frac\{1\}\{\backslash sigma\}\backslash left(1\; +\; \backslash frac\{\backslash xi\; (x-\backslash mu)\}\{\backslash sigma\}\backslash right)^\{\backslash left(-\backslash frac\{1\}\{\backslash xi\}\; -\; 1\backslash right)\}$,

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

The pdf is a solution of the following differential equation: {{Citation needed|date=December 2019}}

:$\backslash left\backslash \{\backslash begin\{array\}\{l\}$

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 $\backslash xi$ and location $\backslash mu$ are both zero, the GPD is equivalent to the exponential distribution.
• With shape $\backslash xi\; =\; -1$, the GPD is equivalent to the continuous uniform distribution $U(0,\; \backslash sigma)$.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 $\backslash xi\; >\; 0$ and location $\backslash mu\; =\; \backslash sigma/\backslash xi$, the GPD is equivalent to the Pareto distribution with scale $x\_m=\backslash sigma/\backslash xi$ and shape $\backslash alpha=1/\backslash xi$.
• If $X$ $\backslash sim$ $GPD$ $($$\backslash mu\; =\; 0$, $\backslash sigma$, $\backslash xi$ $)$, then $Y\; =\; \backslash log\; (X)\; \backslash sim\; exGPD(\backslash sigma,\; \backslash xi)$ [https://www.tandfonline.com/doi/abs/10.1080/03610926.2018.1441418]. (exGPD stands for the exponentiated generalized Pareto distribution.)
• GPD is similar to the Burr distribution.

If U is uniformly distributed on

(0, 1], then

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

and

:$X\; =\; \backslash mu\; -\; \backslash sigma\; \backslash ln(U)\; \backslash sim\; GPD(\backslash mu,\backslash sigma,\backslash xi\; =0).$

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.

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

:$\backslash \; X\backslash \; \backslash vert\backslash \; \backslash Lambda\; \backslash sim\; \backslash operatorname\backslash mathsf\{Exp\}(\backslash Lambda)\backslash$

and

:$\backslash \; \backslash Lambda\; \backslash sim\; \backslash operatorname\backslash mathsf\{Gamma\}(\backslash alpha,\backslash \; \backslash beta)\backslash$

then

:$\backslash \; X\; \backslash sim\; \backslash operatorname\backslash mathsf\{GPD\}(\backslash \; \backslash xi\; =\; 1/\backslash alpha,\backslash \; \backslash sigma\; =\; \backslash beta/\backslash alpha\backslash \; )\backslash$

Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that $\backslash \; \backslash xi\backslash$ 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 $\backslash \; Y\; \backslash sim\; \backslash operatorname\backslash mathsf\{Exponential\}(\backslash \; 1\backslash \; )\backslash$ and $\backslash \; Z\; \backslash sim\; \backslash operatorname\backslash mathsf\{Gamma\}(1/\backslash xi,\backslash \; 1)\backslash \; ,$ we have $\backslash \; \backslash mu\; +\; \backslash frac\{\backslash \; \backslash sigma\backslash \; Y\backslash \; \}\{\backslash \; \backslash xi\backslash \; Z\backslash \; \}\; \backslash sim\; \backslash operatorname\backslash mathsf\{GPD\}(\backslash mu,\backslash \; \backslash sigma,\backslash \; \backslash xi)\; ~.$ This is a consequence of the mixture after setting $\backslash \; \backslash beta\; =\; \backslash alpha\backslash$ and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.

File:ExGPDpdf.png

If $X\; \backslash sim\; GPD$ $($$\backslash mu\; =\; 0$, $\backslash sigma$, $\backslash xi$ $)$, then $Y\; =\; \backslash log\; (X)$ is distributed according to the [https://www.tandfonline.com/doi/abs/10.1080/03610926.2018.1441418 exponentiated generalized Pareto distribution], denoted by $Y$ $\backslash sim$ $exGPD$ $($$\backslash sigma$, $\backslash xi$ $)$.

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

:$g\_\{(\backslash sigma,\; \backslash xi)\}(y)\; =\; \backslash begin\{cases\}\; \backslash frac\{e^y\}\{\backslash sigma\}\backslash bigg(\; 1\; +\; \backslash frac\{\backslash xi\; e^y\}\{\backslash sigma\}\; \backslash bigg)^\{-1/\backslash xi\; -1\}\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash text\{for\; \}\; \backslash xi\; \backslash neq\; 0,\; \backslash \backslash$

\frac{1}{\sigma}e^{y - e^{y}/\sigma} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\, \text{for } \xi = 0 ,\end{cases}

where the support is for $\backslash xi\; \backslash geq\; 0$, and for .

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

The exGPD has finite moments of all orders for all $\backslash sigma>0$ and .

File:Var exGPD.png of the $exGPD(\backslash sigma,\backslash xi)$ as a function of $\backslash xi$. Note that the variance only depends on $\backslash xi$. The red dotted line represents the variance evaluated at $\backslash xi=0$, that is, $\backslash psi\text{'}(1)\; =\; \backslash pi^2/6$.]]

The moment-generating function of $Y\; \backslash sim\; exGPD(\backslash sigma,\backslash xi)$ is

:

\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 $B(a,b)$ and $\backslash Gamma\; (a)$ denote the beta function and gamma function, respectively.

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

:

\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 $\backslash xi\; \backslash in\; (-\backslash infty,\backslash infty)$, the $\backslash log\backslash \; \backslash sigma$ plays as the location parameter under the exponentiated generalized Pareto distribution.

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

:

\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 $\backslash xi$. Note that $\backslash psi\text{'}(1)\; =\; \backslash pi^2/6\; \backslash approx\; 1.644934$.

Note that the roles of the scale parameter $\backslash sigma$ and the shape parameter $\backslash xi$ under $Y\; \backslash sim\; exGPD(\backslash sigma,\; \backslash xi)$ are separably interpretable, which may lead to a robust efficient estimation for the $\backslash xi$ than using the $X\; \backslash sim\; GPD(\backslash sigma,\; \backslash xi)$ [https://www.tandfonline.com/doi/abs/10.1080/03610926.2018.1441418]. The roles of the two parameters are associated each other under $X\; \backslash sim\; GPD(\backslash mu=0,\backslash sigma,\; \backslash xi)$ (at least up to the second central moment); see the formula of variance $Var(X)$ wherein both parameters are participated.

Assume that $X\_\{1:n\}\; =\; (X\_1,\; \backslash cdots,\; X\_n)$ are $n$ observations (need not be i.i.d.) from an unknown heavy-tailed distribution $F$ such that its tail distribution is regularly varying with the tail-index $1/\backslash xi$ (hence, the corresponding shape parameter is $\backslash xi$). To be specific, the tail distribution is described as

:$$

\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 $\backslash xi$, especially when $\backslash xi$ 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 $\backslash xi$: 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 $1\backslash leq\; i\; \backslash leq\; n$, write $X\_\{(i)\}$ for the $i$-th largest value of $X\_1,\; \backslash cdots,\; X\_n$. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [https://books.google.com/books?id=o-clBQAAQBAJ&dq=modeeling+extreme+events+for+insurance&pg=PA1]) based on the $k$ upper order statistics is defined as

:$$

\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 $\backslash widehat\{\backslash xi\}\_\{k\}^\{\backslash text\{Hill\}\}$ at each integer $k\; \backslash in\; \backslash \{\; 2,\; \backslash cdots,\; n\backslash \}$, and then plot the ordered pairs $\backslash \{(k,\backslash widehat\{\backslash xi\}\_\{k\}^\{\backslash text\{Hill\}\})\backslash \}\_\{k=2\}^\{n\}$. Then, select from the set of Hill estimators $\backslash \{\backslash widehat\{\backslash xi\}\_\{k\}^\{\backslash text\{Hill\}\}\backslash \}\_\{k=2\}^\{n\}$ which are roughly constant with respect to $k$: these stable values are regarded as reasonable estimates for the shape parameter $\backslash xi$. If $X\_1,\; \backslash cdots,\; X\_n$ are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter $\backslash xi$ [https://www.jstor.org/stable/1427870].

Note that the Hill estimator $\backslash widehat\{\backslash xi\}\_\{k\}^\{\backslash text\{Hill\}\}$ makes a use of the log-transformation for the observations $X\_\{1:n\}\; =\; (X\_1,\; \backslash cdots,\; X\_n)$. (The Pickand's estimator $\backslash widehat\{\backslash xi\}\_\{k\}^\{\backslash text\{Pickand\}\}$ also employed the log-transformation, but in a slightly different way

[https://www.jstor.org/stable/2242785].)

• Burr distribution
• Pareto distribution
• Generalized extreme value distribution
• [https://www.tandfonline.com/doi/abs/10.1080/03610926.2018.1441418 Exponentiated generalized Pareto distribution]
• Pickands–Balkema–de Haan theorem

{{Reflist|refs}}

• {{Cite journal|last=Pickands|first=James|title=Statistical inference using extreme order statistics|journal=Annals of Statistics|volume=3 s|year=1975|pages=119–131|doi=10.1214/aos/1176343003|doi-access=free|url=https://projecteuclid.org/journals/annals-of-statistics/volume-3/issue-1/Statistical-Inference-Using-Extreme-Order-Statistics/10.1214/aos/1176343003.pdf}}
• {{Cite journal |last1=Balkema |first1=A. |title=Residual life time at great age |journal=Annals of Probability |volume=2 |year=1974 |pages=792–804 |doi=10.1214/aop/1176996548 |first2=Laurens |last2=De Haan |issue=5 |author-link2=Laurens de Haan |doi-access=free }}
• {{Cite journal |last1=Lee|first1=Seyoon |title = Exponentiated generalized Pareto distribution:Properties and applications towards extreme value theory |journal=Communications in Statistics - Theory and Methods|year=2018|pages=1–25 |doi=10.1080/03610926.2018.1441418 |first2=J.H.K. |last2=Kim|volume=48 |issue=8 |arxiv=1708.01686 |s2cid=88514574 }}
• {{cite book|title=Continuous Univariate Distributions Volume 1, second edition|author1=N. L. Johnson |author2=S. Kotz |author3=N. Balakrishnan |publisher=Wiley|location=New York|year=1994|isbn=978-0-471-58495-7}} Chapter 20, Section 12: Generalized Pareto Distributions.
• {{cite book|editor= Duangkamon Chotikapanich|year=2011|title=Modeling Distributions and Lorenz Curves|publisher=Springer|location=New York|author=Barry C. Arnold|chapter=Chapter 7: Pareto and Generalized Pareto Distributions|chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC&pg=PA119|isbn= 9780387727967}}
• {{cite book|author1=Arnold, B. C. |author2=Laguna, L.|year=1977|title= On generalized Pareto distributions with applications to income data|location= Ames, Iowa| publisher=Iowa State University, Department of Economics}}

• [http://www.mathworks.com/help/stats/generalized-pareto-distribution.html Mathworks: Generalized Pareto distribution]

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