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Probability density function  Note: b=0.4, β=3 | |||
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Cumulative distribution function  | |||
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![{\displaystyle =\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d40ede0194fe8297e30eb62598a07b1aaebf91)
![{\displaystyle \left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20459f70785a4f564b5dbb1ee080f91070add07c)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c310906d01a8b2136c968ee71c67056320464d7)
![{\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!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9de7340c6d52753050a5ae813e06912c61335448)
![{\displaystyle =\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ec718073cb46360bb3076cc6daef007b2da181)
![{\displaystyle {\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbecfda266daf8e2d14e0749ef8a6b773b5be24)
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.
Specification[edit]
Probability density function[edit]
The probability density function of the Gamma/Gompertz distribution is:
where is the scale parameter and are the shape parameters of the Gamma/Gompertz distribution.
Cumulative distribution function[edit]
The cumulative distribution function of the Gamma/Gompertz distribution is:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0500f204c3a87c90d1263b07146dd09eb7d199)
Moment generating function[edit]
The moment generating function is given by:
where is a Hypergeometric function.
![{\displaystyle {_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6944910aa72c697ec5ee711df72ca472e877fef5)
Properties[edit]
The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.
Related distributions[edit]
- 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 [1]
- When the shape parameter of a Gompertz distribution varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the distribution of is Gamma/Gompertz.[1]
See also[edit]
Notes[edit]
References[edit]
- Bemmaor, Albert C.; Glady, Nicolas (2012). "Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model". Management Science. 58 (5): 1012–1021. doi:10.1287/mnsc.1110.1461. Archived from the original on 2015-06-26.
- Bemmaor, Albert C.; Glady, Nicolas (2011). "Implementing the Gamma/Gompertz/NBD Model in MATLAB" (PDF). Cergy-Pontoise: ESSEC Business School.[permanent dead link]
- Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society of London. 115: 513–583. doi:10.1098/rstl.1825.0026. JSTOR 107756. S2CID 145157003.
- Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions. Vol. 2 (2nd ed.). New York: John Wiley & Sons. pp. 25–26. ISBN 0-471-58494-0.
- Manton, K. G.; Stallard, E.; Vaupel, J. W. (1986). "Alternative Models for the Heterogeneity of Mortality Risks Among the Aged". Journal of the American Statistical Association. 81 (395): 635–644. doi:10.1080/01621459.1986.10478316. PMID 12155405.
