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The logarithmic gamma distribution (also log gamma distribution ) is a continuous probability distribution . The heavy-tailed distribution is suitable for modeling damage data in the extreme large-scale loss area of industrial, liability and reinsurance.
definition
A constant random variable with the parameters and satisfies the logarithmic gamma distribution if it is the probability density
X
{\ displaystyle X}
a
>
0
{\ displaystyle a> 0}
b
>
0
{\ displaystyle b> 0}
f
(
x
)
=
{
b
a
Γ
(
a
)
x
-
(
b
+
1
)
(
ln
x
)
a
-
1
x
≥
1
0
x
<
1
{\ displaystyle f (x) = {\ begin {cases} {\ dfrac {b ^ {a}} {\ Gamma (a)}} x ^ {- (b + 1)} (\ ln x) ^ {a -1} & x \ geq 1 \\ 0 & x <1 \ end {cases}}}
owns. Your distribution function is then
F.
(
x
)
=
{
γ
(
a
,
b
ln
x
)
Γ
(
a
)
x
≥
1
0
x
<
1
{\ displaystyle F (x) = {\ begin {cases} {\ dfrac {\ gamma (a, b \ ln x)} {\ Gamma (a)}} & x \ geq 1 \\ 0 & x <1 \ end {cases }}}
,
where is the incomplete gamma function .
γ
(
p
,
q
)
{\ displaystyle \ gamma (p, q)}
properties
Expected value
For there is the expectation to
b
>
1
{\ displaystyle b> 1}
E.
(
X
)
=
(
1
-
1
b
)
-
a
{\ displaystyle \ operatorname {E} (X) = \ left (1 - {\ frac {1} {b}} \ right) ^ {- a}}
.
Variance
The variance results for as
b
>
2
{\ displaystyle b> 2}
Var
(
X
)
=
(
1
-
2
b
)
-
a
-
(
1
-
1
b
)
-
2
a
{\ displaystyle \ operatorname {Var} (X) = \ left (1 - {\ frac {2} {b}} \ right) ^ {- a} - \ left (1 - {\ frac {1} {b} } \ right) ^ {- 2a}}
.
Coefficient of variation
The coefficient of variation is obtained immediately from the expected value and the variance
VarK
(
X
)
=
(
1
+
1
b
(
b
-
2
)
)
a
-
1
{\ displaystyle \ operatorname {VarK} (X) = {\ sqrt {\ left (1 + {\ frac {1} {b (b-2)}} \ right) ^ {a} -1}}}
.
Crookedness
The skew can be represented for closed as
b
>
3
{\ displaystyle b> 3}
v
(
X
)
=
(
b
b
-
3
)
a
-
3
(
b
2
(
b
-
2
)
(
b
-
1
)
)
a
+
2
(
b
b
-
1
)
3
a
(
(
b
b
-
2
)
a
-
(
b
b
-
1
)
2
a
)
3
2
{\ displaystyle \ operatorname {v} (X) = {\ frac {\ left ({\ frac {b} {b-3}} \ right) ^ {a} -3 \ left ({\ frac {b ^ { 2}} {(b-2) (b-1)}} \ right) ^ {a} +2 \ left ({\ frac {b} {b-1}} \ right) ^ {3a}} {\ left (\ left ({\ frac {b} {b-2}} \ right) ^ {a} - \ left ({\ frac {b} {b-1}} \ right) ^ {2a} \ right) ^ {\ frac {3} {2}}}}}
.
Moments
Only moments of order exist less than .
b
{\ displaystyle b}
Relationship to other distributions
In actuarial mathematics , the distribution of the number of claims is often modeled with the help of Poisson , negative binomial
or logarithmically distributed random variables . In contrast, the gamma , logarithmic gamma or
logarithmic normal distribution are suitable for describing the amount of damage
.
Relationship to the gamma distribution
If the random variable is gamma distributed , then it is log gamma distributed.
X
{\ displaystyle X}
Y
=
e
X
{\ displaystyle Y = e ^ {X}}
Relationship to Pareto distribution
The Pareto distribution with the parameters and corresponds to the log gamma distribution with the parameters and .
k
{\ displaystyle k}
x
m
i
n
=
1
{\ displaystyle x _ {\ mathrm {min}} = 1}
a
=
1
{\ displaystyle a = 1}
b
=
k
{\ displaystyle b = k}
Individual evidence
↑ Claudia Cottin, Sebastian Döhler: Risk analysis: Modeling, assessment and management of risks with practical examples . Springer-Verlag 2012
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