Gamma/Gompertz distribution

Gamma/Gompertz distribution
	Probability density function; Note: b=0.4, β=3
	Cumulative distribution function
Parameters
Support
PDF
CDF	;
Mean	; ; ; ;
Median
Mode
Variance	; ; ; ; ; ; ; ; ; ;
MGF	; ; ; ;

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

Probability density function

The probability density function of the Gamma/Gompertz distribution is:

f(x;b,s,\beta )={\frac {bse^{bx}\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s+1}}}

where $b>0$ is the scale parameter and $\beta ,s>0\,\!$ are the shape parameters of the Gamma/Gompertz distribution.

Cumulative distribution function

The cumulative distribution function of the Gamma/Gompertz distribution is:

{\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:

{\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 ${_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!$ is a Hypergeometric function.

Properties

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 $b\,\!.$ [1]
When the shape parameter $\eta \,\!$ of a Gompertz distribution varies according to a gamma distribution with shape parameter $\alpha \,\!$ and scale parameter $\beta \,\!$ (mean = $\alpha /\beta \,\!$ ), the distribution of $x$ is Gamma/Gompertz.[1]

Notes

Bemmaor, A.C.; Glady, N. (2012)

References

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.
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. 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[BG-1] Bemmaor, A.C.; Glady, N. (2012)

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Gamma/Gompertz distribution
Probability density function Note: b=0.4, β=3
Cumulative distribution function
Parameters	$b,s,\beta >0\,\!$
Support	$x\in [0,\infty )\!$
PDF	$bse^{bx}\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s+1}{\text{where }}b,s,\beta >0$
CDF	$1-\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s},x>0,b,s,\beta >0$ $1-e^{-bsx},\beta =1$
Mean	$=\left(1/b\right)\left(1/s\right){_{2}{\text{F}}_{1}}\left(s,1;s+1;\left(\beta -1\right)/\beta \right),$ $b,s>0,\beta \neq 1$ $=\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),$ $b>0,s=1,\beta \neq 1$ $=1/\left(bs\right),\quad b,s>0,\beta =1$
Median	$\left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}$
Mode	${\begin{aligned}x^{}&=(1/b)\ln \left[(1/s)(\beta -1)\right],\\&{\text{with }}0<{\text{F}}(x^{})<1-(\beta s)^{s}/\left[(\beta -1)(s+1)\right]^{s}<0.632121,\\&\beta >s+1\\&=0,\quad \beta \leq s+1\\\end{aligned}}$
Variance	$=2(1/b^{2})(1/s^{2})\beta ^{s}{_{3}{\text{F}}_{2}}(s,s,s;s+1,s+1;1-\beta )$ $-{\text{E}}^{2}(\tau \|b,s,\beta ),\quad \beta \neq 1$ $=(1/b^{2})(1/s^{2}),\quad \beta =1$ ${\text{with}}$ ${_{3}{\text{F}}_{2}}(a,b,c;d,e;z)=\sum _{k=0}^{\infty }\{(a)_{k}(b)_{k}(c)_{k}/[(d)_{k}(e)_{k}]\}z^{k}/k!$ ${\text{and}}$ $(a)_{k}=\Gamma (a+k)/\Gamma (a)$
MGF	${\text{E}}(e^{-tx})$ $=\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),$ $\quad \beta \neq 1$ $=sb/(t+sb),\quad \beta =1$ ${\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!$