The gamma-gamma distribution is a univariate distribution for continuous random variables that plays an important role in Bayesian statistics and in inference theory , since it is a mixed distribution .
definition
The probability density function of the gamma-gamma distribution is at
G
G
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α
,
β
,
δ
)
{\ displaystyle Gg (\ alpha, \ beta, \ delta)}
α
,
β
,
δ
>
0
{\ displaystyle \ alpha, \ beta, \ delta> 0}
f
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x
)
=
β
α
B.
(
α
,
δ
)
x
δ
-
1
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β
+
x
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α
+
δ
{\ displaystyle f (x) = {\ frac {\ beta ^ {\ alpha}} {B (\ alpha, \ delta)}} {\ frac {x ^ {\ delta -1}} {(\ beta + x ) ^ {\ alpha + \ delta}}}}
where is Euler's beta function .
B.
(
α
,
β
)
{\ displaystyle B (\ alpha, \ beta)}
properties
Expectation and variance
The expected value is
E.
(
X
)
=
δ
β
α
-
1
{\ displaystyle \ operatorname {E} (X) = {\ frac {\ delta \ beta} {\ alpha -1}}}
, For
α
>
1
{\ displaystyle \ alpha> 1}
and the variance
Var
(
X
)
=
β
2
δ
(
δ
+
α
-
1
)
(
α
-
1
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2
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-
2
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=
E.
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2
1
+
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-
1
δ
α
-
2
{\ displaystyle \ operatorname {Var} (X) = {\ frac {\ beta ^ {2} \ delta (\ delta + \ alpha -1)} {(\ alpha -1) ^ {2} (\ alpha -2 )}} = \ operatorname {E} (X) ^ {2} {\ frac {1 + {\ frac {\ alpha -1} {\ delta}}} {\ alpha -2}}}
, For
α
>
2
{\ displaystyle \ alpha> 2}
mode
The mode is
Mod
(
X
)
=
β
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δ
-
1
)
α
+
1
{\ displaystyle \ operatorname {Mod} (X) = {\ frac {\ beta \ (\ delta -1)} {\ alpha +1}}}
, For
δ
>
1
{\ displaystyle \ delta> 1}
Special case δ = 1
If δ = 1, then is the density function
f
(
x
|
δ
=
1
)
=
α
β
+
x
(
β
β
+
x
)
α
{\ displaystyle f (x | \ delta = 1) = {\ frac {\ alpha} {\ beta + x}} \ left ({\ frac {\ beta} {\ beta + x}} \ right) ^ {\ alpha}}
As you turn this special case of the exponential distribution, with gammaverteiltem parameters .
G
(
1
,
λ
)
=
E.
x
p
(
λ
)
{\ displaystyle G (1, \ lambda) = Exp (\ lambda)}
G
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α
,
β
)
{\ displaystyle G (\ alpha, \ beta)}
λ
{\ displaystyle \ lambda}
Special case β = 1: Inverse beta distribution
A gamma-gamma distribution
corresponds to an inverse beta distribution
G
G
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α
,
β
=
1
,
δ
)
{\ displaystyle Gg (\ alpha, \ beta = 1, \ delta)}
I.
n
v
B.
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a
=
δ
,
b
=
α
)
{\ displaystyle {\ mathcal {InvB}} (a = \ delta, b = \ alpha)}
Relationship to the gamma distribution
If the second parameter of the gamma distribution is a random variable that is distributed like a gamma distribution, then the resulting random variable is distributed like a gamma-gamma distribution.
ϵ
{\ displaystyle \ epsilon}
G
(
d
,
ϵ
)
{\ displaystyle G (d, \ epsilon)}
G
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a
,
b
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{\ displaystyle G (a, b)}
G
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a
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b
,
d
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{\ displaystyle G (a, b, d)}
Relationship to the exponential distribution
If the parameter of the exponential distribution is a random variable that is distributed like a gamma distribution, then the resulting random variable is distributed like a gamma-gamma distribution.
λ
{\ displaystyle \ lambda}
E.
x
p
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λ
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{\ displaystyle Exp (\ lambda)}
G
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a
,
b
)
{\ displaystyle G (a, b)}
G
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a
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b
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1
)
{\ displaystyle G (a, b, 1)}
literature
Leonhard Held: Methods of statistical inference. Likelihood and Bayes , with the assistance of Daniel Sabanés Bové, Spektrum Akademischer Verlag Heidelberg 2008, ISBN 978-3-8274-1939-2
See also
Discrete univariate distributions
Continuous univariate distributions
Multivariate distributions
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