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

## Usage

```
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)
```

## Arguments

- 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 \(P(X \le x)\), 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

`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.

## Details

If \(w \sim Gamma(k,1)\), 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

$$f(x | a, b, k) = \frac{b}{\Gamma(k)} \frac{x^{bk - 1}}{a^{bk}} $$$$ \exp(-(x/a)^b)$$

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.

## References

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.