Английская Википедия:Generalized Pareto distribution
Шаблон:Short description Шаблон:About Шаблон:More citations needed Шаблон:Probability distribution{(1-3\xi)}\,\;(\xi<1/3)</math>
| kurtosis =<math>\frac{3(1-2\xi)(2\xi^2+\xi+3)}{(1-3\xi)(1-4\xi)}-3\,\;(\xi<1/4)</math>
| mgf =<math>e^{\theta\mu}\,\sum_{j=0}^\infty \left[\frac{(\theta\sigma)^j}{\prod_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)</math>|
| cf =<math>e^{it\mu}\,\sum_{j=0}^\infty \left[\frac{(it\sigma)^j}{\prod_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)</math>
| variance =<math>\frac{\sigma^2}{(1-\xi)^2(1-2\xi)}\, \; (\xi < 1/2)</math>
| moments =<math>\xi = \frac{1}{2}\left(1 - \frac{(E[X] - \mu)^2}{V[X]}\right)</math>
<math> \sigma = (E[X] - \mu)(1 - \xi)</math>
| ES =<math>\begin{cases}\mu + \sigma\left[ \frac{(1-p)^{-\xi} }{1-\xi} + \frac{(1-p)^{-\xi} -1 }{\xi} \right]&,\xi \neq 0\\\mu + \sigma[1- \ln(1-p) ]&,\xi =0\end{cases}</math>[1]
| bPOE =<math>\begin{cases}\frac{ \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{- \frac{1}{\xi} } }{(1-\xi)^{ \frac{1}{\xi} } } &,\xi \neq 0\\\ e^{ 1 - \left( \frac{x-\mu}{\sigma} \right) }&,\xi =0\end{cases}</math>[1]
}}
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 <math>\mu</math>, scale <math>\sigma</math>, and shape <math>\xi</math>.[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 <math> \kappa = - \xi \,</math>.[5]
Definition
The standard cumulative distribution function (cdf) of the GPD is defined by[6]
- <math>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} </math>
where the support is <math> z \geq 0 </math> for <math> \xi \geq 0</math> and <math> 0 \leq z \leq - 1 /\xi </math> for <math> \xi < 0</math>. The corresponding probability density function (pdf) is
- <math>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} </math>
Characterization
The related location-scale family of distributions is obtained by replacing the argument z by <math>\frac{x-\mu}{\sigma}</math> and adjusting the support accordingly.
The cumulative distribution function of <math>X \sim GPD(\mu, \sigma, \xi)</math> (<math>\mu\in\mathbb R</math>, <math>\sigma>0</math>, and <math>\xi\in\mathbb R</math>) is
- <math>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} </math> where the support of <math>X</math> is <math> x \geqslant \mu </math> when <math> \xi \geqslant 0 \,</math>, and <math> \mu \leqslant x \leqslant \mu - \sigma /\xi </math> when <math> \xi < 0</math>.
The probability density function (pdf) of <math>X \sim GPD(\mu, \sigma, \xi)</math> is
- <math>f_{(\mu,\sigma,\xi)}(x) = \frac{1}{\sigma}\left(1 + \frac{\xi (x-\mu)}{\sigma}\right)^{\left(-\frac{1}{\xi} - 1\right)}</math>,
again, for <math> x \geqslant \mu </math> when <math> \xi \geqslant 0</math>, and <math> \mu \leqslant x \leqslant \mu - \sigma /\xi </math> when <math> \xi < 0</math>.
The pdf is a solution of the following differential equation: Шаблон:Citation needed
- <math>\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\} </math>
Special cases
- If the shape <math>\xi</math> and location <math>\mu</math> are both zero, the GPD is equivalent to the exponential distribution.
- With shape <math>\xi = -1</math>, the GPD is equivalent to the continuous uniform distribution <math>U(0, \sigma)</math>.[7]
- With shape <math>\xi > 0</math> and location <math>\mu = \sigma/\xi</math>, the GPD is equivalent to the Pareto distribution with scale <math>x_m=\sigma/\xi</math> and shape <math>\alpha=1/\xi</math>.
- If <math> X </math> <math>\sim</math> <math>GPD</math> <math>(</math><math>\mu = 0</math>, <math>\sigma</math>, <math>\xi</math> <math>)</math>, then <math> Y = \log (X) \sim exGPD(\sigma, \xi)</math> [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
- <math> X = \mu + \frac{\sigma (U^{-\xi}-1)}{\xi} \sim GPD(\mu, \sigma, \xi \neq 0)</math>
and
- <math> X = \mu - \sigma \ln(U) \sim GPD(\mu,\sigma,\xi =0).</math>
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.
- <math>X|\Lambda \sim \operatorname{Exp}(\Lambda) </math>
and
- <math>\Lambda \sim \operatorname{Gamma}(\alpha, \beta) </math>
then
- <math>X \sim \operatorname{GPD}(\xi = 1/\alpha, \ \sigma = \beta/\alpha) </math>
Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:<math>\xi</math> 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 <math>Y \sim \text{Exponential}(1)</math> and <math>Z \sim \text{Gamma}(1/\xi, 1)</math>, we have <math>\mu + \sigma \frac{Y}{\xi Z} \sim \text{GPD}(\mu,\sigma,\xi)</math>. This is a consequence of the mixture after setting <math>\beta=\alpha</math> and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.
Exponentiated generalized Pareto distribution
The exponentiated generalized Pareto distribution (exGPD)
If <math> X \sim GPD</math> <math>(</math><math>\mu = 0</math>, <math>\sigma</math>, <math>\xi</math> <math>)</math>, then <math> Y = \log (X)</math> is distributed according to the exponentiated generalized Pareto distribution, denoted by <math> Y</math> <math>\sim</math> <math>exGPD</math> <math>(</math><math>\sigma</math>, <math>\xi</math> <math>)</math>.
The probability density function(pdf) of <math> Y </math> <math>\sim</math> <math>exGPD</math> <math>(</math><math>\sigma</math>, <math>\xi</math> <math>)\,\, (\sigma >0) </math> is
- <math> 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}</math>
where the support is <math> -\infty < y < \infty </math> for <math> \xi \geq 0 </math>, and <math> -\infty < y \leq \log(-\sigma/\xi)</math> for <math> \xi < 0 </math>.
For all <math>\xi</math>, the <math>\log \sigma </math> becomes the location parameter. See the right panel for the pdf when the shape <math>\xi</math> is positive.
The exGPD has finite moments of all orders for all <math>\sigma>0</math> and <math>-\infty< \xi < \infty </math>.
The moment-generating function of <math> Y \sim exGPD(\sigma,\xi)</math> is
- <math> 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}</math>
where <math>B(a,b) </math> and <math> \Gamma (a) </math> denote the beta function and gamma function, respectively.
The expected value of <math> Y </math> <math>\sim</math> <math>exGPD</math> <math>(</math><math>\sigma</math>, <math>\xi</math> <math>)</math> depends on the scale <math> \sigma</math> and shape <math> \xi </math> parameters, while the <math> \xi </math> participates through the digamma function:
- <math> 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}</math>
Note that for a fixed value for the <math> \xi \in (-\infty,\infty) </math>, the <math> \log\ \sigma </math> plays as the location parameter under the exponentiated generalized Pareto distribution.
The variance of <math> Y </math> <math>\sim</math> <math>exGPD</math> <math>(</math><math>\sigma</math>, <math>\xi</math> <math>)</math> depends on the shape parameter <math> \xi </math> only through the polygamma function of order 1 (also called the trigamma function):
- <math> 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}</math>
See the right panel for the variance as a function of <math>\xi</math>. Note that <math> \psi'(1) = \pi^2/6 \approx 1.644934 </math>.
Note that the roles of the scale parameter <math>\sigma</math> and the shape parameter <math>\xi</math> under <math>Y \sim exGPD(\sigma, \xi)</math> are separably interpretable, which may lead to a robust efficient estimation for the <math>\xi</math> than using the <math>X \sim GPD(\sigma, \xi)</math> [2]. The roles of the two parameters are associated each other under <math>X \sim GPD(\mu=0,\sigma, \xi)</math> (at least up to the second central moment); see the formula of variance <math>Var(X)</math> wherein both parameters are participated.
The Hill's estimator
Assume that <math> X_{1:n} = (X_1, \cdots, X_n) </math> are <math>n</math> observations (not need to be i.i.d.) from an unknown heavy-tailed distribution <math> F </math> such that its tail distribution is regularly varying with the tail-index <math>1/\xi </math> (hence, the corresponding shape parameter is <math>\xi </math>). To be specific, the tail distribution is described as
- <math>
\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.}
</math>
It is of a particular interest in the extreme value theory to estimate the shape parameter <math>\xi</math>, especially when <math>\xi</math> is positive (so called the heavy-tailed distribution).
Let <math>F_u</math> 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 <math>F</math>, and large <math>u</math>, <math>F_u</math> is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate <math>\xi</math>: 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 <math> 1\leq i \leq n </math>, write <math> X_{(i)} </math> for the <math>i</math>-th largest value of <math> X_1, \cdots, X_n </math>. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [3]) based on the <math>k</math> upper order statistics is defined as
- <math>
\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.
</math>
In practice, the Hill estimator is used as follows. First, calculate the estimator <math>\widehat{\xi}_{k}^{\text{Hill}}</math> at each integer <math>k \in \{ 2, \cdots, n\}</math>, and then plot the ordered pairs <math>\{(k,\widehat{\xi}_{k}^{\text{Hill}})\}_{k=2}^{n}</math>. Then, select from the set of Hill estimators <math>\{\widehat{\xi}_{k}^{\text{Hill}}\}_{k=2}^{n}</math> which are roughly constant with respect to <math>k</math>: these stable values are regarded as reasonable estimates for the shape parameter <math>\xi</math>. If <math> X_1, \cdots, X_n </math> are i.i.d., then the Hill's estimator is a consistent estimator for the shape parameter <math>\xi</math> [4].
Note that the Hill estimator <math>\widehat{\xi}_{k}^{\text{Hill}}</math> makes a use of the log-transformation for the observations <math> X_{1:n} = (X_1, \cdots, X_n) </math>. (The Pickand's estimator <math>\widehat{\xi}_{k}^{\text{Pickand}}</math> also employed the log-transformation, but in a slightly different way [5].)
See also
- Burr distribution
- Pareto distribution
- Generalized extreme value distribution
- Exponentiated generalized Pareto distribution
- Pickands–Balkema–de Haan theorem
References
Further reading
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book Chapter 20, Section 12: Generalized Pareto Distributions.
- Шаблон:Cite book
- Шаблон:Cite book
External links
- ↑ 1,0 1,1 Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Castillo, Enrique, and Ali S. Hadi. "Fitting the generalized Pareto distribution to data." Journal of the American Statistical Association 92.440 (1997): 1609-1620.
