GenGamma.orig | R Documentation |

Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the original parameterisation from Stacy (1962).

dgengamma.orig(x, shape, scale = 1, k, log = FALSE) pgengamma.orig(q, shape, scale = 1, k, lower.tail = TRUE, log.p = FALSE) Hgengamma.orig(x, shape, scale = 1, k) hgengamma.orig(x, shape, scale = 1, k) qgengamma.orig(p, shape, scale = 1, k, lower.tail = TRUE, log.p = FALSE) rgengamma.orig(n, shape, scale = 1, k)

`x, q` |
vector of quantiles. |

`shape` |
vector of “Weibull” shape parameters. |

`scale` |
vector of scale parameters. |

`k` |
vector of “Gamma” shape 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 |

If *w ~ Gamma(k, 1)*, then *x = exp(w/shape + log(scale))*
follows the original generalised gamma distribution with the
parameterisation given here (Stacy 1962). Defining
`shape`

*=b>0*, `scale`

*=a>0*, *x* has
probability density

* f(x
| a, b, k) = (b / Γ(k)) (x^{bk -1} / a^{bk}) exp(-(x/a)^b)*

* f(x | a, b, k) = (b / Γ(k)) (x^{bk -1} / a^{bk})
exp(-(x/a)^b)*

The original generalized gamma distribution simplifies to the gamma, exponential and Weibull distributions with the following parameterisations:

`dgengamma.orig(x, shape, scale, k=1)` | `=`
| `dweibull(x, shape, scale)` |

```
dgengamma.orig(x,
shape=1, scale, k)
``` | `=` | ```
dgamma(x, shape=k,
scale)
``` |

`dgengamma.orig(x, shape=1, scale, k=1)` | `=`
| `dexp(x, rate=1/scale)` |

Also as k tends to infinity, it tends to the log normal (as in
`dlnorm`

) with the following parameters (Lawless,
1980):

```
dlnorm(x, meanlog=log(scale) + log(k)/shape,
sdlog=1/(shape*sqrt(k)))
```

For more stable behaviour as the distribution tends to the log-normal, an
alternative parameterisation was developed by Prentice (1974). This is
given in `dgengamma`

, and is now preferred for statistical
modelling. It is also more flexible, including a further new class of
distributions with negative shape `k`

.

The generalized F distribution `GenF.orig`

, and its similar
alternative parameterisation `GenF`

, extend the generalized
gamma to four parameters.

`dgengamma.orig`

gives the density, `pgengamma.orig`

gives the distribution function, `qgengamma.orig`

gives the quantile
function, `rgengamma.orig`

generates random deviates,
`Hgengamma.orig`

retuns the cumulative hazard and
`hgengamma.orig`

the hazard.

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

Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:1187-92.

Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539-544.

Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409-419.

`GenGamma`

, `GenF.orig`

,
`GenF`

, `Lognormal`

, `GammaDist`

,
`Weibull`

.

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