Generalized Pareto distribution
family of probability distributions often used to model tails or extreme values
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
μ
{\displaystyle \mu }
, scale
σ
{\displaystyle \sigma }
, and shape
ξ
{\displaystyle \xi }
. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as
κ
=
−
ξ
{\displaystyle \kappa =-\xi \,}
. This parameterization was introduced by James Pickands III .
Definition
The cumulative distribution function of
X
∼
GPD
(
μ
,
σ
,
ξ
)
{\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )}
(
μ
∈
R
{\displaystyle \mu \in \mathbb {R} }
,
σ
>
0
{\displaystyle \sigma >0}
, and
ξ
∈
R
{\displaystyle \xi \in \mathbb {R} }
) is
F
μ
,
σ
,
ξ
(
x
)
=
{
1
−
(
1
+
ξ
x
−
μ
σ
)
−
1
/
ξ
for
ξ
≠
0
,
1
−
exp
(
−
x
−
μ
σ
)
for
ξ
=
0
,
{\displaystyle F_{\mu ,\sigma ,\xi }(x)={\begin{cases}1-\left(1+\xi {\frac {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
≥
μ
{\displaystyle x\geq \mu }
when
ξ
≥
0
{\displaystyle \xi \geq 0}
, and
μ
≤
x
≤
μ
−
σ
/
ξ
{\displaystyle \mu \leq x\leq \mu -\sigma /\xi }
when
ξ
<
0
{\displaystyle \xi <0}
.
The probability density function (pdf) of
X
∼
GPD
(
μ
,
σ
,
ξ
)
{\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )}
is
f
μ
,
σ
,
ξ
(
x
)
=
{
1
σ
(
1
+
ξ
x
−
μ
σ
)
−
1
−
1
/
ξ
for
ξ
≠
0
,
1
σ
exp
(
−
x
−
μ
σ
)
for
ξ
=
0
,
{\displaystyle f_{\mu ,\sigma ,\xi }(x)={\begin{cases}{\frac {1}{\sigma }}\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-1-1/\xi }&{\text{for }}\xi \neq 0,\\{\frac {1}{\sigma }}\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0,\end{cases}}}
again, for
x
≥
μ
{\displaystyle x\geq \mu }
when
ξ
≥
0
{\displaystyle \xi \geq 0}
, and
μ
≤
x
≤
μ
−
σ
/
ξ
{\displaystyle \mu \leq x\leq \mu -\sigma /\xi }
when
ξ
<
0
{\displaystyle \xi <0}
.
The GPD survival function (sf),
F
¯
(
x
)
=
1
−
F
(
x
)
{\displaystyle {\bar {F}}(x)=1-F(x)}
, is the solution to the following nonlinear ordinary differential equation:
F
¯
′
(
x
)
+
1
σ
F
¯
(
x
)
1
+
ξ
=
0
,
F
¯
(
μ
)
=
1
{\displaystyle {\bar {F}}'(x)+{\frac {1}{\sigma }}{\bar {F}}(x)^{1+\xi }=0,\qquad {\bar {F}}(\mu )=1}
.
Thus, the tail shape
ξ
{\displaystyle \xi }
quantifies the nonlinear deviation from linear systems and the scale
σ
{\displaystyle \sigma }
is associated with the linear source of uncertainty.
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