Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the parameterisation originating from Prentice (1974). Also known as the (generalized) log-gamma distribution.

## Usage

```
dgengamma(x, mu = 0, sigma = 1, Q, log = FALSE)
pgengamma(q, mu = 0, sigma = 1, Q, lower.tail = TRUE, log.p = FALSE)
Hgengamma(x, mu = 0, sigma = 1, Q)
hgengamma(x, mu = 0, sigma = 1, Q)
qgengamma(p, mu = 0, sigma = 1, Q, lower.tail = TRUE, log.p = FALSE)
rgengamma(n, mu = 0, sigma = 1, Q)
```

## Arguments

- x, q
vector of quantiles.

- mu
Vector of ``location'' parameters.

- sigma
Vector of ``scale'' parameters. Note the inconsistent meanings of the term ``scale'' - this parameter is analogous to the (log-scale) standard deviation of the log-normal distribution, ``sdlog'' in

`dlnorm`

, rather than the ``scale'' parameter of the gamma distribution`dgamma`

. Constrained to be positive.- Q
Vector of 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`

gives the density, `pgengamma`

gives the
distribution function, `qgengamma`

gives the quantile function,
`rgengamma`

generates random deviates, `Hgengamma`

retuns the
cumulative hazard and `hgengamma`

the hazard.

## Details

If \(\gamma \sim Gamma(Q^{-2}, 1)\) , and \(w = log(Q^2 \gamma) / Q\), then \(x = \exp(\mu + \sigma w)\) follows the generalized gamma distribution with probability density function

$$f(x | \mu, \sigma, Q) = \frac{|Q|(Q^{-2})^{Q^{-2}}}{\sigma x \Gamma(Q^{-2})} \exp(Q^{-2}(Qw - \exp(Qw)))$$

This parameterisation is preferred to the original
parameterisation of the generalized gamma by Stacy (1962) since it
is more numerically stable near to \(Q=0\) (the log-normal
distribution), and allows \(Q<=0\). The original is available
in this package as `dgengamma.orig`

, for the sake of
completion and compatibility with other software - this is
implicitly restricted to `Q`

>0 (or `k`

>0 in the original
notation). The parameters of `dgengamma`

and
`dgengamma.orig`

are related as follows.

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

```
dgengamma(x, mu=log(scale) + log(k)/shape, sigma=1/(shape*sqrt(k)),
Q=1/sqrt(k))
```

The generalized gamma distribution simplifies to the gamma, log-normal and Weibull distributions with the following parameterisations:

`dgengamma(x, mu, sigma, Q=0)` | `=` | `dlnorm(x, mu, sigma)` |

`dgengamma(x, mu, sigma, Q=1)` | `=` | `dweibull(x, shape=1/sigma, scale=exp(mu))` |

`dgengamma(x, mu, sigma, Q=sigma)` | `=` | ```
dgamma(x,
shape=1/sigma^2, rate=exp(-mu) / sigma^2)
``` |

The properties of the generalized gamma and its applications to survival analysis are discussed in detail by Cox (2007).

The generalized F distribution `GenF`

extends the generalized
gamma to four parameters.

## References

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

Farewell, V. T. and Prentice, R. L. (1977). A study of distributional shape in life testing. Technometrics 19(1):69-75.

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

Cox, C., Chu, H., Schneider, M. F. and Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine 26:4252-4374

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