DescTools (version 0.99.60)

Gompertz: The Gompertz distribution

Description

Density, distribution function, 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)

Value

dGompertz gives the density, pGompertz gives the distribution function, qGompertz gives the quantile 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 <[e-mail]>

Details

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

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

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 $1-\exp (b/a)$ 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.

See Also

dexp