Shifted Gompertz distribution
Density, distribution function, and random generation for the shifted Gompertz distribution.
dsgomp(x, b, eta, log = FALSE) psgomp(q, b, eta, lower.tail = TRUE, log.p = FALSE) rsgomp(n, b, eta)
x, q |
vector of quantiles. |
b, eta |
positive valued scale and shape parameters; both need to be positive. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
n |
number of observations. If |
If X follows exponential distribution parametrized by scale b and
Y follows reparametrized Gumbel distribution with cumulative distribution function
F(x) = exp(-η*exp(-b*x)) parametrized by
scale b and shape η, then max(X,Y) follows shifted
Gompertz distribution parametrized by scale b>0 and shape η>0.
The above relation is used by rsgomp function for random generation from
shifted Gompertz distribution.
Probability density function
f(x) = b*exp(-b*x) * exp(-η*exp(-b*x)) * (1 + η*(1 - exp(-b*x)))
Cumulative distribution function
F(x) = (1-exp(-b*x)) * exp(-η*exp(-b*x))
Bemmaor, A.C. (1994). Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity. [In:] G. Laurent, G.L. Lilien & B. Pras. Research Traditions in Marketing. Boston: Kluwer Academic Publishers. pp. 201-223.
Jimenez, T.F. and Jodra, P. (2009). A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution. Communications in Statistics - Theory and Methods, 38(1), 78-89.
Jimenez T.F. (2014). Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods. Journal of Computational and Applied Mathematics, 255(1), 867-877.
x <- rsgomp(1e5, 0.4, 1) hist(x, 50, freq = FALSE) curve(dsgomp(x, 0.4, 1), 0, 30, col = "red", add = TRUE) hist(psgomp(x, 0.4, 1)) plot(ecdf(x)) curve(psgomp(x, 0.4, 1), 0, 30, col = "red", lwd = 2, add = TRUE)
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