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

Details

The Gompertz distribution with shape parameter \(a\) and rate parameter \(b\) has probability density function

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

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