In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks. Quick Facts Parameters, Support ...Gamma/Gompertz distribution Probability density functionNote: b=0.4, β=3 Cumulative distribution functionParameters b , s , β > 0 {\displaystyle b,s,\beta >0\,\!} Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )\!} PDF b s e b x β s / ( β − 1 + e b x ) s + 1 where b , s , β > 0 {\displaystyle bse^{bx}\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s+1}{\text{where }}b,s,\beta >0} CDF 1 − β s / ( β − 1 + e b x ) s , x > 0 , b , s , β > 0 {\displaystyle 1-\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s},x>0,b,s,\beta >0} 1 − e − b s x , β = 1 {\displaystyle 1-e^{-bsx},\beta =1} Mean = ( 1 / b ) ( 1 / s ) 2 F 1 ( s , 1 ; s + 1 ; ( β − 1 ) / β ) , {\displaystyle =\left(1/b\right)\left(1/s\right){_{2}{\text{F}}_{1}}\left(s,1;s+1;\left(\beta -1\right)/\beta \right),} b , s > 0 , β ≠ 1 {\displaystyle b,s>0,\beta \neq 1} = ( 1 / b ) [ β / ( β − 1 ) ] ln ( β ) , {\displaystyle =\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),} b > 0 , s = 1 , β ≠ 1 {\displaystyle b>0,s=1,\beta \neq 1} = 1 / ( b s ) , b , s > 0 , β = 1 {\displaystyle =1/\left(bs\right),\quad b,s>0,\beta =1} Median ( 1 / b ) ln { β [ ( 1 / 2 ) − 1 / s − 1 ] + 1 } {\displaystyle \left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}} Mode x ∗ = ( 1 / b ) ln [ ( 1 / s ) ( β − 1 ) ] , with 0 < F ( x ∗ ) < 1 − ( β s ) s / [ ( β − 1 ) ( s + 1 ) ] s < 0.632121 , β > s + 1 = 0 , β ≤ s + 1 {\displaystyle {\begin{aligned}x^{*}&=(1/b)\ln \left[(1/s)(\beta -1)\right],\\&{\text{with }}0<{\text{F}}(x^{*})<1-(\beta s)^{s}/\left[(\beta -1)(s+1)\right]^{s}<0.632121,\\&\beta >s+1\\&=0,\quad \beta \leq s+1\\\end{aligned}}} Variance = 2 ( 1 / b 2 ) ( 1 / s 2 ) β s 3 F 2 ( s , s , s ; s + 1 , s + 1 ; 1 − β ) {\displaystyle =2(1/b^{2})(1/s^{2})\beta ^{s}{_{3}{\text{F}}_{2}}(s,s,s;s+1,s+1;1-\beta )} − E 2 ( τ | b , s , β ) , β ≠ 1 {\displaystyle -{\text{E}}^{2}(\tau |b,s,\beta ),\quad \beta \neq 1} = ( 1 / b 2 ) ( 1 / s 2 ) , β = 1 {\displaystyle =(1/b^{2})(1/s^{2}),\quad \beta =1} with {\displaystyle {\text{with}}} 3 F 2 ( a , b , c ; d , e ; z ) = ∑ k = 0 ∞ { ( a ) k ( b ) k ( c ) k / [ ( d ) k ( e ) k ] } z k / k ! {\displaystyle {_{3}{\text{F}}_{2}}(a,b,c;d,e;z)=\sum _{k=0}^{\infty }\{(a)_{k}(b)_{k}(c)_{k}/[(d)_{k}(e)_{k}]\}z^{k}/k!} and {\displaystyle {\text{and}}} ( a ) k = Γ ( a + k ) / Γ ( a ) {\displaystyle (a)_{k}=\Gamma (a+k)/\Gamma (a)} MGF E ( e − t x ) {\displaystyle {\text{E}}(e^{-tx})} = β s [ s b / ( t + s b ) ] 2 F 1 ( s + 1 , ( t / b ) + s ; ( t / b ) + s + 1 ; 1 − β ) , {\displaystyle =\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),} β ≠ 1 {\displaystyle \quad \beta \neq 1} = s b / ( t + s b ) , β = 1 {\displaystyle =sb/(t+sb),\quad \beta =1} with 2 F 1 ( a , b ; c ; z ) = ∑ k = 0 ∞ [ ( a ) k ( b ) k / ( c ) k ] z k / k ! {\displaystyle {\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!} Close Remove adsSpecificationSummarizePerspective Probability density function The probability density function of the Gamma/Gompertz distribution is: f ( x ; b , s , β ) = b s e b x β s ( β − 1 + e b x ) s + 1 {\displaystyle f(x;b,s,\beta )={\frac {bse^{bx}\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s+1}}}} where b > 0 {\displaystyle b>0} is the scale parameter and β , s > 0 {\displaystyle \beta ,s>0\,\!} are the shape parameters of the Gamma/Gompertz distribution. Cumulative distribution function The cumulative distribution function of the Gamma/Gompertz distribution is: F ( x ; b , s , β ) = 1 − β s ( β − 1 + e b x ) s , x > 0 , b , s , β > 0 = 1 − e − b s x , β = 1 {\displaystyle {\begin{aligned}F(x;b,s,\beta )&=1-{\frac {\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s}}},{\ }x>0,{\ }b,s,\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}} Moment generating function The moment generating function is given by: E ( e − t x ) = { β s s b t + s b 2 F 1 ( s + 1 , ( t / b ) + s ; ( t / b ) + s + 1 ; 1 − β ) , β ≠ 1 ; s b t + s b , β = 1. {\displaystyle {\begin{aligned}{\text{E}}(e^{-tx})={\begin{cases}\displaystyle \beta ^{s}{\frac {sb}{t+sb}}{\ }{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),&\beta \neq 1;\\\displaystyle {\frac {sb}{t+sb}},&\beta =1.\end{cases}}\end{aligned}}} where 2 F 1 ( a , b ; c ; z ) = ∑ k = 0 ∞ [ ( a ) k ( b ) k / ( c ) k ] z k / k ! {\displaystyle {_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!} is a Hypergeometric function. Remove adsProperties The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left. Related distributions When β = 1, this reduces to an Exponential distribution with parameter sb. The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter b . {\displaystyle b\,\!.} [1] When the shape parameter η {\displaystyle \eta \,\!} of a Gompertz distribution varies according to a gamma distribution with shape parameter α {\displaystyle \alpha \,\!} and scale parameter β {\displaystyle \beta \,\!} (mean = α / β {\displaystyle \alpha /\beta \,\!} ), the distribution of x {\displaystyle x} is Gamma/Gompertz.[1] See also Gompertz distribution Customer lifetime value NotesLoading content...ReferencesLoading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads
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