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 ${\displaystyle \mu }${\displaystyle \mu }, scale ${\displaystyle \sigma }${\displaystyle \sigma }, and shape ${\displaystyle \xi }${\displaystyle \xi }.^{[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 ${\displaystyle \kappa =-\xi ,円}${\displaystyle \kappa =-\xi ,円}.^[5]

With shape ${\displaystyle \xi >0}${\displaystyle \xi >0} and location ${\displaystyle \mu =\sigma /\xi }${\displaystyle \mu =\sigma /\xi }, the GPD is equivalent to the Pareto distribution with scale ${\displaystyle x_{m}=\sigma /\xi }${\displaystyle x_{m}=\sigma /\xi } and shape ${\displaystyle \alpha =1/\xi }${\displaystyle \alpha =1/\xi }.

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

$${\displaystyle F_{(\mu ,\sigma ,\xi )}(x)={\begin{cases}1-\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\1円-\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0,\end{cases}}}$${\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円-\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0,\end{cases}}} where the support of ${\displaystyle X}${\displaystyle X} is ${\displaystyle x\geq \mu }${\displaystyle x\geq \mu } when ${\displaystyle \xi \geq 0,円}${\displaystyle \xi \geq 0,円}, and ${\displaystyle \mu \leq x\leq \mu -\sigma /\xi }${\displaystyle \mu \leq x\leq \mu -\sigma /\xi } when ${\displaystyle \xi <0}${\displaystyle \xi <0}.

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

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

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

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

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

The standard cumulative distribution function (cdf) of the GPD is defined using ${\displaystyle z={\frac {x-\mu }{\sigma }}.}${\displaystyle z={\frac {x-\mu }{\sigma }}.}^[6]

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

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

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

• If ${\displaystyle \xi =0}${\displaystyle \xi =0}, the GPD is the exponential distribution.
• If ${\displaystyle \xi >0}${\displaystyle \xi >0}, the GPD is the Pareto distribution with shape ${\displaystyle \alpha =1/\xi }${\displaystyle \alpha =1/\xi }.
• If ${\displaystyle \xi <0}${\displaystyle \xi <0}, the GPD is the power function distribution with shape ${\displaystyle \alpha =-1/\xi }${\displaystyle \alpha =-1/\xi }.
• If ${\displaystyle \xi =-1}${\displaystyle \xi =-1}, the GPD is the continuous uniform distribution ${\displaystyle U(0,\sigma )}${\displaystyle U(0,\sigma )}.^[7]
• If ${\displaystyle X\sim \mathrm {GPD} (\mu =0,\sigma ,\xi )}${\displaystyle X\sim \mathrm {GPD} (\mu =0,\sigma ,\xi )}, then ${\displaystyle Y=\log(X)\sim \mathrm {exGPD} (\sigma ,\xi )}${\displaystyle Y=\log(X)\sim \mathrm {exGPD} (\sigma ,\xi )} [1]. (exGPD stands for the exponentiated generalized Pareto distribution.)
• GPD is similar to the Burr distribution.

• It is often of interest to predict probabilities of out-of-sample data under the assumption that both the training data and the out-of-sample data follow a GPD.
• Predictions of probabilities generated by substituting maximum likelihood estimates of the GPD parameters into the cumulative distribution function ignore parameter uncertainty. As a result, the probabilities are not well calibrated, do not reflect the frequencies of out-of-sample events, and, in particular, underestimate the probabilities of out-of-sample tail events.^[8]
• Predictions generated using the objective Bayesian approach of calibrating prior prediction have been shown to greatly reduce this underestimation, although not completely eliminate it.^[8] Calibrating prior prediction is implemented in the R software package fitdistcp.[2]

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

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

Both formulas are obtained by inversion of the cdf.

The Pareto package in R and the gprnd command in the Matlab Statistics Toolbox can be used 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.

$${\displaystyle X\mid \Lambda \sim \mathrm {Exp} (\Lambda )}$${\displaystyle X\mid \Lambda \sim \mathrm {Exp} (\Lambda )} and $${\displaystyle \Lambda \sim \mathrm {Gamma} (\alpha ,,円\beta )}$${\displaystyle \Lambda \sim \mathrm {Gamma} (\alpha ,,円\beta )} then $${\displaystyle X\sim \mathrm {GPD} (\xi =1/\alpha ,\ \sigma =\beta /\alpha )}$${\displaystyle X\sim \mathrm {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 ${\displaystyle \xi }${\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 ${\displaystyle Y\sim \mathrm {Exp} (1)}${\displaystyle Y\sim \mathrm {Exp} (1)} and ${\displaystyle Z\sim \mathrm {Gamma} (1/\xi ,,1円),,円}${\displaystyle Z\sim \mathrm {Gamma} (1/\xi ,,1円),,円} we have ${\displaystyle \mu +{\frac {\sigma Y}{\xi Z}}\sim \mathrm {GPD} (\mu ,\sigma ,\xi ),円.}${\displaystyle \mu +{\frac {\sigma Y}{\xi Z}}\sim \mathrm {GPD} (\mu ,\sigma ,\xi ),円.} This is a consequence of the mixture after setting ${\displaystyle \beta =\alpha }${\displaystyle \beta =\alpha } and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.

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

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

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

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

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

The moment-generating function of ${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )}${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )} is $${\displaystyle M_{Y}(s)=\operatorname {E} \left[e^{sY}\right]={\begin{cases}-{\frac {1}{\xi }}\left(-{\frac {\sigma }{\xi }}\right)^{s}B(s{+}1,,円-1/\xi ),&{\text{for }}&-1<s<\infty ,&\xi <0,\\[1ex]{\frac {1}{\xi }}\left({\frac {\sigma }{\xi }}\right)^{s}B(s{+}1,,1円/\xi -s)&{\text{for }}&-1<s<1/\xi ,&\xi >0,\\[1ex]\sigma ^{s}\Gamma (1+s),&{\text{for }}&-1<s<\infty ,&\xi =0,\end{cases}}}$${\displaystyle M_{Y}(s)=\operatorname {E} \left[e^{sY}\right]={\begin{cases}-{\frac {1}{\xi }}\left(-{\frac {\sigma }{\xi }}\right)^{s}B(s{+}1,,円-1/\xi ),&{\text{for }}&-1<s<\infty ,&\xi <0,\\[1ex]{\frac {1}{\xi }}\left({\frac {\sigma }{\xi }}\right)^{s}B(s{+}1,,1円/\xi -s)&{\text{for }}&-1<s<1/\xi ,&\xi >0,\\[1ex]\sigma ^{s}\Gamma (1+s),&{\text{for }}&-1<s<\infty ,&\xi =0,\end{cases}}} where ${\displaystyle B(a,b)}${\displaystyle B(a,b)} and ${\displaystyle \Gamma (a)}${\displaystyle \Gamma (a)} denote the beta function and gamma function, respectively.

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

The variance of ${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )}${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )} depends on the shape parameter ${\displaystyle \xi }${\displaystyle \xi } only through the polygamma function of order 1 (also called the trigamma function): $${\displaystyle \operatorname {Var} [Y]={\begin{cases}\psi '(1)-\psi '(-1/\xi +1)&{\text{for }}\xi <0,\\\psi '(1)+\psi '(1/\xi )&{\text{for }}\xi >0,\\\psi '(1)&{\text{for }}\xi =0.\end{cases}}}$${\displaystyle \operatorname {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 ${\displaystyle \xi }${\displaystyle \xi }. Note that ${\displaystyle \psi '(1)=\pi ^{2}/6\approx 1.644934}${\displaystyle \psi '(1)=\pi ^{2}/6\approx 1.644934}.

Note that the roles of the scale parameter ${\displaystyle \sigma }${\displaystyle \sigma } and the shape parameter ${\displaystyle \xi }${\displaystyle \xi } under ${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )}${\displaystyle Y\sim \mathrm {exGPD} (\sigma ,\xi )} are separably interpretable, which may lead to a robust efficient estimation for the ${\displaystyle \xi }${\displaystyle \xi } than using the ${\displaystyle X\sim \mathrm {GPD} (\sigma ,\xi )}${\displaystyle X\sim \mathrm {GPD} (\sigma ,\xi )} [3]. The roles of the two parameters are associated each other under ${\displaystyle X\sim \mathrm {GPD} (\mu =0,\sigma ,\xi )}${\displaystyle X\sim \mathrm {GPD} (\mu =0,\sigma ,\xi )} (at least up to the second central moment); see the formula of variance ${\displaystyle Var(X)}${\displaystyle Var(X)} wherein both parameters are participated.

Assume that ${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})} are ${\displaystyle n}${\displaystyle n} observations (need not be i.i.d.) from an unknown heavy-tailed distribution ${\displaystyle F}${\displaystyle F} such that its tail distribution is regularly varying with the tail-index ${\displaystyle 1/\xi }${\displaystyle 1/\xi } (hence, the corresponding shape parameter is ${\displaystyle \xi }${\displaystyle \xi }). To be specific, the tail distribution is described as $${\displaystyle {\bar {F}}(x)=1-F(x)=L(x)\cdot x^{-1/\xi },,円,円,円,円,円{\text{for some }}\xi >0,,円,円{\text{where }}L{\text{ is a slowly varying function.}}}$${\displaystyle {\bar {F}}(x)=1-F(x)=L(x)\cdot x^{-1/\xi },,円,円,円,円,円{\text{for some }}\xi >0,,円,円{\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 ${\displaystyle \xi }${\displaystyle \xi }, especially when ${\displaystyle \xi }${\displaystyle \xi } is positive (so called the heavy-tailed distribution).

Let ${\displaystyle F_{u}}${\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 ${\displaystyle F}${\displaystyle F}, and large ${\displaystyle u}${\displaystyle u}, ${\displaystyle F_{u}}${\displaystyle F_{u}} is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate ${\displaystyle \xi }${\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 ${\displaystyle 1\leq i\leq n}${\displaystyle 1\leq i\leq n}, write ${\displaystyle X_{(i)}}${\displaystyle X_{(i)}} for the ${\displaystyle i}${\displaystyle i}-th largest value of ${\displaystyle X_{1},\cdots ,X_{n}}${\displaystyle X_{1},\cdots ,X_{n}}. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [4]) based on the ${\displaystyle k}${\displaystyle k} upper order statistics is defined as $${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}={\widehat {\xi }}_{k}^{\text{Hill}}(X_{1:n})={\frac {1}{k-1}}\sum _{j=1}^{k-1}\log {\bigg (}{\frac {X_{(j)}}{X_{(k)}}}{\bigg )},,円,円,円,円,円,円,円,円{\text{for }}2\leq k\leq n.}$${\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 )},,円,円,円,円,円,円,円,円{\text{for }}2\leq k\leq n.} In practice, the Hill estimator is used as follows. First, calculate the estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}} at each integer ${\displaystyle k\in \{2,\cdots ,n\}}${\displaystyle k\in \{2,\cdots ,n\}}, and then plot the ordered pairs ${\displaystyle \{(k,{\widehat {\xi }}_{k}^{\text{Hill}})\}_{k=2}^{n}}${\displaystyle \{(k,{\widehat {\xi }}_{k}^{\text{Hill}})\}_{k=2}^{n}}. Then, select from the set of Hill estimators ${\displaystyle \{{\widehat {\xi }}_{k}^{\text{Hill}}\}_{k=2}^{n}}${\displaystyle \{{\widehat {\xi }}_{k}^{\text{Hill}}\}_{k=2}^{n}} which are roughly constant with respect to ${\displaystyle k}${\displaystyle k}: these stable values are regarded as reasonable estimates for the shape parameter ${\displaystyle \xi }${\displaystyle \xi }. If ${\displaystyle X_{1},\cdots ,X_{n}}${\displaystyle X_{1},\cdots ,X_{n}} are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter ${\displaystyle \xi }${\displaystyle \xi } [5].

Note that the Hill estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}${\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}} makes a use of the log-transformation for the observations ${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}${\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}. (The Pickand's estimator ${\displaystyle {\widehat {\xi }}_{k}^{\text{Pickand}}}${\displaystyle {\widehat {\xi }}_{k}^{\text{Pickand}}} also employed the log-transformation, but in a slightly different way [6].)

• 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

• Continuous distributions
• Power laws
• Probability distributions with non-finite variance

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