Gamma-gamma distribution

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 ${\displaystyle Gg(\alpha ,\beta ,\delta )}$ ${\displaystyle \alpha ,\beta ,\delta >0}$

${\displaystyle f(x)={\frac {\beta ^{\alpha }}{B(\alpha ,\delta )}}{\frac {x^{\delta -1}}{(\beta +x)^{\alpha +\delta }}}}$

where is Euler's beta function . ${\displaystyle B(\alpha ,\beta )}$

properties

Expectation and variance

The expected value is

${\displaystyle \operatorname {E} (X)={\frac {\delta \beta }{\alpha -1}}}$, For ${\displaystyle \alpha >1}$

and the variance

${\displaystyle \operatorname {Var} (X)={\frac {\beta ^{2}\delta (\delta +\alpha -1)}{(\alpha -1)^{2}(\alpha -2)}}=\operatorname {E} (X)^{2}{\frac {1+{\frac {\alpha -1}{\delta }}}{\alpha -2}}}$, For ${\displaystyle \alpha >2}$

mode

The mode is

${\displaystyle \operatorname {Mod} (X)={\frac {\beta \ (\delta -1)}{\alpha +1}}}$, For ${\displaystyle \delta >1}$

Special case δ = 1

If δ = 1, then is the density function

${\displaystyle f(x|\delta =1)={\frac {\alpha }{\beta +x}}\left({\frac {\beta }{\beta +x}}\right)^{\alpha }}$

As you turn this special case of the exponential distribution, with gammaverteiltem parameters . ${\displaystyle G(1,\lambda )=Exp(\lambda )}$ ${\displaystyle G(\alpha ,\beta )}$ ${\displaystyle \lambda }$

Special case β = 1: Inverse beta distribution

A gamma-gamma distribution corresponds to an inverse beta distribution ${\displaystyle Gg(\alpha ,\beta =1,\delta )}$ ${\displaystyle {\mathcal {InvB}}(a=\delta ,b=\alpha )}$

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. ${\displaystyle \epsilon }$ ${\displaystyle G(d,\epsilon )}$ ${\displaystyle G(a,b)}$ ${\displaystyle G(a,b,d)}$

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. ${\displaystyle \lambda }$ ${\displaystyle Exp(\lambda )}$ ${\displaystyle G(a,b)}$ ${\displaystyle G(a,b,1)}$

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

• Exponential distribution