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1
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G e n e r a t i n g g e n e r a l i z e d P a r e t o r a n d o m v a r i a b l e s
T o g g l e G e n e r a t i n g g e n e r a l i z e d P a r e t o r a n d o m v a r i a b l e s s u b s e c t i o n
4 . 1
G e n e r a t i n g G P D r a n d o m v a r i a b l e s
4 . 2
G P D a s a n E x p o n e n t i a l - G a m m a M i x t u r e
5
E x p o n e n t i a t e d g e n e r a l i z e d P a r e t o d i s t r i b u t i o n
T o g g l e E x p o n e n t i a t e d g e n e r a l i z e d P a r e t o d i s t r i b u t i o n s u b s e c t i o n
5 . 1
T h e e x p o n e n t i a t e d g e n e r a l i z e d P a r e t o d i s t r i b u t i o n ( e x G P D )
6
T h e H i l l ' s e s t i m a t o r
7
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8
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9
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10
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
G e n e r a l i z e d P a r e t o d i s t r i b u t i o n
2 l a n g u a g e s
● C a t a l à
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● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Family of probability distributions often used to model tails or extreme values
This article is about a particular family of continuous distributions referred to as the generalized Pareto distribution. For the hierarchy of generalized Pareto distributions, see
Pareto distribution .
Generalized Pareto distribution
Probability density function
GPD distribution functions for
μ
=
0
{\displaystyle \mu =0}
and different values of
σ
{\displaystyle \sigma }
and
ξ
{\displaystyle \xi }
Cumulative distribution function
Parameters
μ
∈
(
−
∞
,
∞
)
{\displaystyle \mu \in (-\infty ,\infty )\,}
location (real )
σ
∈
(
0
,
∞
)
{\displaystyle \sigma \in (0,\infty )\,}
scale (real)
ξ
∈
(
−
∞
,
∞
)
{\displaystyle \xi \in (-\infty ,\infty )\,}
shape (real) Support
x
⩾
μ
(
ξ
⩾
0
)
{\displaystyle x\geqslant \mu \,\;(\xi \geqslant 0)}
μ
⩽
x
⩽
μ
−
σ
/
ξ
(
ξ
<
0
)
{\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi \,\;(\xi <0)}
PDF
(
1
+
ξ
z
)
−
(
1
/
ξ
+
1
)
{\displaystyle (1+\xi z)^{-(1/\xi +1)}}
where
z
=
x
−
μ
σ
{\displaystyle z={\frac {x-\mu }{\sigma }}}
CDF
1
−
(
1
+
ξ
z
)
−
1
/
ξ
{\displaystyle 1-(1+\xi z)^{-1/\xi }\,}
Mean
μ
+
σ
1
−
ξ
(
ξ
<
1
)
{\displaystyle \mu +{\frac {\sigma }{1-\xi }}\,\;(\xi <1 )}
Median
μ
+
σ
(
2
ξ
−
1
)
ξ
{\displaystyle \mu +{\frac {\sigma (2^{\xi }-1)}{\xi }}}
Mode
μ
{\displaystyle \mu }
Variance
σ
2
(
1
−
ξ
)
2
(
1
−
2
ξ
)
(
ξ
<
1
/
2
)
{\displaystyle {\frac {\sigma ^{2}}{(1-\xi )^{2}(1-2\xi )}}\,\;(\xi <1/2)}
Skewness
2
(
1
+
ξ
)
1
−
2
ξ
(
1
−
3
ξ
)
(
ξ
<
1
/
3
)
{\displaystyle {\frac {2(1+\xi ){\sqrt {1-2\xi }}}{(1-3\xi )}}\,\;(\xi <1/3)}
Excess kurtosis
3
(
1
−
2
ξ
)
(
2
ξ
2
+
ξ
+
3
)
(
1
−
3
ξ
)
(
1
−
4
ξ
)
−
3
(
ξ
<
1
/
4
)
{\displaystyle {\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)}
Entropy
log
(
σ
)
+
ξ
+
1
{\displaystyle \log(\sigma )+\xi +1}
MGF
e
θ
μ
∑
j
=
0
∞
[
(
θ
σ
)
j
∏
k
=
0
j
(
1
−
k
ξ
)
]
,
(
k
ξ
<
1
)
{\displaystyle e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1 )}
CF
e
i
t
μ
∑
j
=
0
∞
[
(
i
t
σ
)
j
∏
k
=
0
j
(
1
−
k
ξ
)
]
,
(
k
ξ
<
1
)
{\displaystyle e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1 )}
Method of moments
ξ
=
1
2
(
1
−
(
E
[
X
]
−
μ
)
2
V
[
X
]
)
{\displaystyle \xi ={\frac {1}{2}}\left(1-{\frac {(E[X ]-\mu )^{2}}{V[X ]}}\right)}
σ
=
(
E
[
X
]
−
μ
)
(
1
−
ξ
)
{\displaystyle \sigma =(E[X ]-\mu )(1-\xi )}
Expected shortfall
{
μ
+
σ
[
(
1
−
p
)
−
ξ
1
−
ξ
+
(
1
−
p
)
−
ξ
−
1
ξ
]
,
ξ
≠
0
μ
+
σ
[
1
−
ln
(
1
−
p
)
]
,
ξ
=
0
{\displaystyle {\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}}}
[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
μ
{\displaystyle \mu }
, scale
σ
{\displaystyle \sigma }
, and shape
ξ
{\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
κ
=
−
ξ
{\displaystyle \kappa =-\xi \,}
.[5]
Definition
[ edit ]
The standard cumulative distribution function (cdf) of the GPD is defined by[6]
F
ξ
(
z
)
=
{
1
−
(
1
+
ξ
z
)
−
1
/
ξ
for
ξ
≠
0
,
1
−
e
−
z
for
ξ
=
0.
{\displaystyle 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}}}
where the support is
z
≥
0
{\displaystyle z\geq 0}
for
ξ
≥
0
{\displaystyle \xi \geq 0}
and
0
≤
z
≤
−
1
/
ξ
{\displaystyle 0\leq z\leq -1/\xi }
for
ξ
<
0
{\displaystyle \xi <0}
. The corresponding probability density function (pdf) is
f
ξ
(
z
)
=
{
(
1
+
ξ
z
)
−
ξ
+
1
ξ
for
ξ
≠
0
,
e
−
z
for
ξ
=
0.
{\displaystyle 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}}}
Characterization
[ edit ]
The related location-scale family of distributions is obtained by replacing the argument z by
x
−
μ
σ
{\displaystyle {\frac {x-\mu }{\sigma }}}
and adjusting the support accordingly.
The cumulative distribution function of
X
∼
G
P
D
(
μ
,
σ
,
ξ
)
{\displaystyle X\sim 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+{\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
{\displaystyle X}
is
x
⩾
μ
{\displaystyle x\geqslant \mu }
when
ξ
⩾
0
{\displaystyle \xi \geqslant 0\,}
, and
μ
⩽
x
⩽
μ
−
σ
/
ξ
{\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi }
when
ξ
<
0
{\displaystyle \xi <0}
.
The probability density function (pdf) of
X
∼
G
P
D
(
μ
,
σ
,
ξ
)
{\displaystyle X\sim GPD(\mu ,\sigma ,\xi )}
is
f
(
μ
,
σ
,
ξ
)
(
x
)
=
1
σ
(
1
+
ξ
(
x
−
μ
)
σ
)
(
−
1
ξ
−
1
)
{\displaystyle f_{(\mu ,\sigma ,\xi )}(x )={\frac {1}{\sigma }}\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{\left(-{\frac {1}{\xi }}-1\right)}}
,
again, for
x
⩾
μ
{\displaystyle x\geqslant \mu }
when
ξ
⩾
0
{\displaystyle \xi \geqslant 0}
, and
μ
⩽
x
⩽
μ
−
σ
/
ξ
{\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi }
when
ξ
<
0
{\displaystyle \xi <0}
.
The pdf is a solution of the following differential equation : [citation needed ]
{
f
′
(
x
)
(
−
μ
ξ
+
σ
+
ξ
x
)
+
(
ξ
+
1
)
f
(
x
)
=
0
,
f
(
0
)
=
(
1
−
μ
ξ
σ
)
−
1
ξ
−
1
σ
}
{\displaystyle \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\}}
Special cases
[ edit ]
If the shape
ξ
{\displaystyle \xi }
and location
μ
{\displaystyle \mu }
are both zero, the GPD is equivalent to the exponential distribution .
With shape
ξ
=
−
1
{\displaystyle \xi =-1}
, the GPD is equivalent to the continuous uniform distribution
U
(
0
,
σ
)
{\displaystyle U(0,\sigma )}
.[7]
With shape
ξ
>
0
{\displaystyle \xi >0}
and location
μ
=
σ
/
ξ
{\displaystyle \mu =\sigma /\xi }
, the GPD is equivalent to the Pareto distribution with scale
x
m
=
σ
/
ξ
{\displaystyle x_{m}=\sigma /\xi }
and shape
α
=
1
/
ξ
{\displaystyle \alpha =1/\xi }
.
If
X
{\displaystyle X}
∼
{\displaystyle \sim }
G
P
D
{\displaystyle GPD}
(
{\displaystyle (}
μ
=
0
{\displaystyle \mu =0}
,
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
{\displaystyle )}
, then
Y
=
log
(
X
)
∼
e
x
G
P
D
(
σ
,
ξ
)
{\displaystyle Y=\log(X )\sim exGPD(\sigma ,\xi )}
[1 ] . (exGPD stands for the exponentiated generalized Pareto distribution .)
GPD is similar to the Burr distribution .
Generating generalized Pareto random variables
[ edit ]
Generating GPD random variables
[ edit ]
If U is uniformly distributed on
(0, 1 ], then
X
=
μ
+
σ
(
U
−
ξ
−
1
)
ξ
∼
G
P
D
(
μ
,
σ
,
ξ
≠
0
)
{\displaystyle X=\mu +{\frac {\sigma (U^{-\xi }-1)}{\xi }}\sim GPD(\mu ,\sigma ,\xi \neq 0)}
and
X
=
μ
−
σ
ln
(
U
)
∼
G
P
D
(
μ
,
σ
,
ξ
=
0
)
.
{\displaystyle 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
[ edit ]
A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter.
X
|
Λ
∼
Exp
(
Λ
)
{\displaystyle X|\Lambda \sim \operatorname {Exp} (\Lambda )}
and
Λ
∼
Gamma
(
α
,
β
)
{\displaystyle \Lambda \sim \operatorname {Gamma} (\alpha ,\beta )}
then
X
∼
GPD
(
ξ
=
1
/
α
,
σ
=
β
/
α
)
{\displaystyle X\sim \operatorname {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:
ξ
{\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
Y
∼
Exponential
(
1
)
{\displaystyle Y\sim {\text{Exponential}}(1 )}
and
Z
∼
Gamma
(
1
/
ξ
,
1
)
{\displaystyle Z\sim {\text{Gamma}}(1/\xi ,1)}
, we have
μ
+
σ
Y
ξ
Z
∼
GPD
(
μ
,
σ
,
ξ
)
{\displaystyle \mu +\sigma {\frac {Y}{\xi Z}}\sim {\text{GPD}}(\mu ,\sigma ,\xi )}
. This is a consequence of the mixture after setting
β
=
α
{\displaystyle \beta =\alpha }
and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.
Exponentiated generalized Pareto distribution
[ edit ]
The exponentiated generalized Pareto distribution (exGPD)
[ edit ]
The pdf of the
e
x
G
P
D
(
σ
,
ξ
)
{\displaystyle exGPD(\sigma ,\xi )}
(exponentiated generalized Pareto distribution) for different values
σ
{\displaystyle \sigma }
and
ξ
{\displaystyle \xi }
.
If
X
∼
G
P
D
{\displaystyle X\sim GPD}
(
{\displaystyle (}
μ
=
0
{\displaystyle \mu =0}
,
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
{\displaystyle )}
, then
Y
=
log
(
X
)
{\displaystyle Y=\log(X )}
is distributed according to the exponentiated generalized Pareto distribution , denoted by
Y
{\displaystyle Y}
∼
{\displaystyle \sim }
e
x
G
P
D
{\displaystyle exGPD}
(
{\displaystyle (}
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
{\displaystyle )}
.
The probability density function (pdf) of
Y
{\displaystyle Y}
∼
{\displaystyle \sim }
e
x
G
P
D
{\displaystyle exGPD}
(
{\displaystyle (}
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
(
σ
>
0
)
{\displaystyle )\,\,(\sigma >0)}
is
g
(
σ
,
ξ
)
(
y
)
=
{
e
y
σ
(
1
+
ξ
e
y
σ
)
−
1
/
ξ
−
1
for
ξ
≠
0
,
1
σ
e
y
−
e
y
/
σ
for
ξ
=
0
,
{\displaystyle 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}}}
where the support is
−
∞
<
y
<
∞
{\displaystyle -\infty <y<\infty }
for
ξ
≥
0
{\displaystyle \xi \geq 0}
, and
−
∞
<
y
≤
log
(
−
σ
/
ξ
)
{\displaystyle -\infty <y\leq \log(-\sigma /\xi )}
for
ξ
<
0
{\displaystyle \xi <0}
.
For all
ξ
{\displaystyle \xi }
, the
log
σ
{\displaystyle \log \sigma }
becomes the location parameter. See the right panel for the pdf when the shape
ξ
{\displaystyle \xi }
is positive.
The exGPD has finite moments of all orders for all
σ
>
0
{\displaystyle \sigma >0}
and
−
∞
<
ξ
<
∞
{\displaystyle -\infty <\xi <\infty }
.
The variance of the
e
x
G
P
D
(
σ
,
ξ
)
{\displaystyle exGPD(\sigma ,\xi )}
as a function of
ξ
{\displaystyle \xi }
. Note that the variance only depends on
ξ
{\displaystyle \xi }
. The red dotted line represents the variance evaluated at
ξ
=
0
{\displaystyle \xi =0}
, that is,
ψ
′
(
1
)
=
π
2
/
6
{\displaystyle \psi '(1 )=\pi ^{2}/6}
.
The moment-generating function of
Y
∼
e
x
G
P
D
(
σ
,
ξ
)
{\displaystyle Y\sim exGPD(\sigma ,\xi )}
is
M
Y
(
s
)
=
E
[
e
s
Y
]
=
{
−
1
ξ
(
−
σ
ξ
)
s
B
(
s
+
1
,
−
1
/
ξ
)
for
s
∈
(
−
1
,
∞
)
,
ξ
<
0
,
1
ξ
(
σ
ξ
)
s
B
(
s
+
1
,
1
/
ξ
−
s
)
for
s
∈
(
−
1
,
1
/
ξ
)
,
ξ
>
0
,
σ
s
Γ
(
1
+
s
)
for
s
∈
(
−
1
,
∞
)
,
ξ
=
0
,
{\displaystyle 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}}}
where
B
(
a
,
b
)
{\displaystyle B(a,b)}
and
Γ
(
a
)
{\displaystyle \Gamma (a )}
denote the beta function and gamma function , respectively.
The expected value of
Y
{\displaystyle Y}
∼
{\displaystyle \sim }
e
x
G
P
D
{\displaystyle exGPD}
(
{\displaystyle (}
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
{\displaystyle )}
depends on the scale
σ
{\displaystyle \sigma }
and shape
ξ
{\displaystyle \xi }
parameters, while the
ξ
{\displaystyle \xi }
participates through the digamma function :
E
[
Y
]
=
{
log
(
−
σ
ξ
)
+
ψ
(
1
)
−
ψ
(
−
1
/
ξ
+
1
)
for
ξ
<
0
,
log
(
σ
ξ
)
+
ψ
(
1
)
−
ψ
(
1
/
ξ
)
for
ξ
>
0
,
log
σ
+
ψ
(
1
)
for
ξ
=
0.
{\displaystyle 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}}}
Note that for a fixed value for the
ξ
∈
(
−
∞
,
∞
)
{\displaystyle \xi \in (-\infty ,\infty )}
, the
log
σ
{\displaystyle \log \ \sigma }
plays as the location parameter under the exponentiated generalized Pareto distribution.
The variance of
Y
{\displaystyle Y}
∼
{\displaystyle \sim }
e
x
G
P
D
{\displaystyle exGPD}
(
{\displaystyle (}
σ
{\displaystyle \sigma }
,
ξ
{\displaystyle \xi }
)
{\displaystyle )}
depends on the shape parameter
ξ
{\displaystyle \xi }
only through the polygamma function of order 1 (also called the trigamma function ):
V
a
r
[
Y
]
=
{
ψ
′
(
1
)
−
ψ
′
(
−
1
/
ξ
+
1
)
for
ξ
<
0
,
ψ
′
(
1
)
+
ψ
′
(
1
/
ξ
)
for
ξ
>
0
,
ψ
′
(
1
)
for
ξ
=
0.
{\displaystyle 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
ξ
{\displaystyle \xi }
. Note that
ψ
′
(
1
)
=
π
2
/
6
≈
1.644934
{\displaystyle \psi '(1 )=\pi ^{2}/6\approx 1.644934}
.
Note that the roles of the scale parameter
σ
{\displaystyle \sigma }
and the shape parameter
ξ
{\displaystyle \xi }
under
Y
∼
e
x
G
P
D
(
σ
,
ξ
)
{\displaystyle Y\sim exGPD(\sigma ,\xi )}
are separably interpretable, which may lead to a robust efficient estimation for the
ξ
{\displaystyle \xi }
than using the
X
∼
G
P
D
(
σ
,
ξ
)
{\displaystyle X\sim GPD(\sigma ,\xi )}
[2 ] . The roles of the two parameters are associated each other under
X
∼
G
P
D
(
μ
=
0
,
σ
,
ξ
)
{\displaystyle X\sim GPD(\mu =0,\sigma ,\xi )}
(at least up to the second central moment); see the formula of variance
V
a
r
(
X
)
{\displaystyle Var(X )}
wherein both parameters are participated.
The Hill's estimator
[ edit ]
Assume that
X
1
:
n
=
(
X
1
,
⋯
,
X
n
)
{\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}
are
n
{\displaystyle n}
observations (not need to be i.i.d.) from an unknown heavy-tailed distribution
F
{\displaystyle F}
such that its tail distribution is regularly varying with the tail-index
1
/
ξ
{\displaystyle 1/\xi }
(hence, the corresponding shape parameter is
ξ
{\displaystyle \xi }
). To be specific, the tail distribution is described as
F
¯
(
x
)
=
1
−
F
(
x
)
=
L
(
x
)
⋅
x
−
1
/
ξ
,
for some
ξ
>
0
,
where
L
is a slowly varying function.
{\displaystyle {\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
ξ
{\displaystyle \xi }
, especially when
ξ
{\displaystyle \xi }
is positive (so called the heavy-tailed distribution).
Let
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
F
{\displaystyle F}
, and large
u
{\displaystyle u}
,
F
u
{\displaystyle F_{u}}
is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate
ξ
{\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
1
≤
i
≤
n
{\displaystyle 1\leq i\leq n}
, write
X
(
i
)
{\displaystyle X_{(i )}}
for the
i
{\displaystyle i}
-th largest value of
X
1
,
⋯
,
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 [3 ] ) based on the
k
{\displaystyle k}
upper order statistics is defined as
ξ
^
k
Hill
=
ξ
^
k
Hill
(
X
1
:
n
)
=
1
k
−
1
∑
j
=
1
k
−
1
log
(
X
(
j
)
X
(
k
)
)
,
for
2
≤
k
≤
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 )},\,\,\,\,\,\,\,\,{\text{for }}2\leq k\leq n.}
In practice, the Hill estimator is used as follows. First, calculate the estimator
ξ
^
k
Hill
{\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}
at each integer
k
∈
{
2
,
⋯
,
n
}
{\displaystyle k\in \{2,\cdots ,n\}}
, and then plot the ordered pairs
{
(
k
,
ξ
^
k
Hill
)
}
k
=
2
n
{\displaystyle \{(k,{\widehat {\xi }}_{k}^{\text{Hill}})\}_{k=2}^{n}}
. Then, select from the set of Hill estimators
{
ξ
^
k
Hill
}
k
=
2
n
{\displaystyle \{{\widehat {\xi }}_{k}^{\text{Hill}}\}_{k=2}^{n}}
which are roughly constant with respect to
k
{\displaystyle k}
: these stable values are regarded as reasonable estimates for the shape parameter
ξ
{\displaystyle \xi }
. If
X
1
,
⋯
,
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
ξ
{\displaystyle \xi }
[4 ] .
Note that the Hill estimator
ξ
^
k
Hill
{\displaystyle {\widehat {\xi }}_{k}^{\text{Hill}}}
makes a use of the log-transformation for the observations
X
1
:
n
=
(
X
1
,
⋯
,
X
n
)
{\displaystyle X_{1:n}=(X_{1},\cdots ,X_{n})}
. (The Pickand's estimator
ξ
^
k
Pickand
{\displaystyle {\widehat {\xi }}_{k}^{\text{Pickand}}}
also employed the log-transformation, but in a slightly different way
[5 ] .)
See also
[ edit ]
References
[ edit ]
^ Coles, Stuart (2001-12-12). An Introduction to Statistical Modeling of Extreme Values . Springer. p. 75. ISBN 9781852334598 .
^ Dargahi-Noubary, G. R. (1989). "On tail estimation: An improved method". Mathematical Geology . 21 (8 ): 829–842. Bibcode :1989MatGe..21..829D . doi :10.1007/BF00894450 . S2CID 122710961 .
^ Hosking, J. R. M.; Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distribution". Technometrics . 29 (3 ): 339–349. doi :10.2307/1269343 . JSTOR 1269343 .
^ Davison, A. C. (1984-09-30). "Modelling Excesses over High Thresholds, with an Application" . In de Oliveira, J. Tiago (ed.). Statistical Extremes and Applications . Kluwer. p. 462. ISBN 9789027718044 .
^ Embrechts, Paul; Klüppelberg, Claudia ; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance . Springer. p. 162. ISBN 9783540609315 .
^ Castillo, Enrique, and Ali S. Hadi. "Fitting the generalized Pareto distribution to data." Journal of the American Statistical Association 92.440 (1997): 1609-1620.
Further reading
[ edit ]
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.
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Generalized_Pareto_distribution&oldid=1233344725 "
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