flexsurv (version 1.1.1)

# 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)`

## 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.

## 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.

## 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.

`dexp`