Gompertz | R Documentation |

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

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)

`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` |
vector of probabilities. |

`n` |
number of observations. If |

The Gompertz distribution with `shape`

parameter *a* and
`rate`

parameter *b* has probability density function

*f(x | a, b) = b exp(ax)
exp(-b/a (exp(ax) - 1))*

and hazard

*h(x | a, b) = b exp(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 (exp(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`

.

`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.

Some implementations of the Gompertz restrict *a* to be strictly
positive, which ensures that the probability of survival decreases to zero
as *x* increases to infinity. The more flexible implementation given
here is consistent with `streg`

in Stata.

The functions `eha::dgompertz`

and similar available in the
package eha label the parameters the other way round, so that what is
called the `shape`

there is called the `rate`

here, and what is
called `1 / scale`

there is called the `shape`

here. The
terminology here is consistent with the exponential `dexp`

and
Weibull `dweibull`

distributions in R.

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

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

`dexp`

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