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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
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f
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{\ 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
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{\ 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
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{\ displaystyle \ gamma (p, q)}
properties Expected value For there is the expectation to
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E.
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{\ displaystyle \ operatorname {E} (X) = \ left (1 - {\ frac {1} {b}} \ right) ^ {- a}}
Variance The variance results for as
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Var
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{\ 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
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{\ displaystyle \ operatorname {VarK} (X) = {\ sqrt {\ left (1 + {\ frac {1} {b (b-2)}} \ right) ^ {a} -1}}}
Crookedness The skew can be represented for closed as
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v
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{\ 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.
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{\ displaystyle X}
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{\ 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 .
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Individual evidence
↑ Claudia Cottin, Sebastian Döhler: Risk analysis: Modeling, assessment and management of risks with practical examples . Springer-Verlag 2012
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