Gompertz: The Gompertz distribution

Description

Density, distribution function, hazards, quantile function and random generation for the Gompertz distribution with unrestricted shape.

Usage

dgompertz(x, shape, rate = 1, log = FALSE)
pgompertz(q, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
qgompertz(p, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
rgompertz(n, shape = 1, rate = 1)
hgompertz(x, shape, rate = 1, log = FALSE)
Hgompertz(x, shape, rate = 1, log = FALSE)

Value

dgompertz gives the density, pgompertz gives the distribution function, qgompertz gives the quantile function, hgompertz gives the hazard function, Hgompertz gives the cumulative hazard function, and rgompertz generates random deviates.

Arguments

x, q: vector of quantiles.
shape, rate: vector of shape and rate parameters.
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, $P(X > x)$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Author

Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>

Details

The Gompertz distribution with shape parameter $a$ and rate parameter $b$ has probability density function

$$f(x | a, b) = be^{ax}\exp(-b/a (e^{ax} - 1))$$

and hazard

$$h(x | a, b) = b e^{ax}$$

The hazard is increasing for shape $a>0$ and decreasing for $a<0$. For $a=0$ the Gompertz is equivalent to the exponential distribution with constant hazard and rate $b$.

The probability distribution function is $$F(x | a, b) = 1 - \exp(-b/a (e^{ax} - 1))$$

Thus if $a$ is negative, letting $x$ tend to infinity shows that there is a non-zero probability $\exp(b/a)$ of living forever. On these occasions qgompertz and rgompertz will return Inf.

References

Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.