ShiftGomp: Shifted Gompertz distribution

In extraDistr: Additional Univariate and Multivariate Distributions

Shifted Gompertz distribution

Description

Density, distribution function, and random generation for the shifted Gompertz distribution.

Usage

dsgomp(x, b, eta, log = FALSE)

psgomp(q, b, eta, lower.tail = TRUE, log.p = FALSE)

rsgomp(n, b, eta)

Arguments

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 \le x] otherwise, P[X > x].
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If X follows exponential distribution parametrized by scale b and Y follows reparametrized Gumbel distribution with cumulative distribution function F(x) = \exp(-\eta e^{-bx}) parametrized by scale b and shape \eta, then \max(X,Y) follows shifted Gompertz distribution parametrized by scale b>0 and shape \eta>0. The above relation is used by rsgomp function for random generation from shifted Gompertz distribution.

Probability density function

f(x) = b e^{-bx} \exp(-\eta e^{-bx}) \left[1 + \eta(1 - e^{-bx})\right]

Cumulative distribution function

F(x) = (1-e^{-bx}) \exp(-\eta e^{-bx})

References

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.

Examples

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)

extraDistr documentation built on May 29, 2024, 9:31 a.m.