# GenGamma.orig: Generalized gamma distribution (original parameterisation) In flexsurv: Flexible parametric survival models

## Description

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

## Usage

 ```1 2 3 4 5 6``` ``` dgengamma.orig(x, shape, scale=1, k, log = FALSE) pgengamma.orig(q, shape, scale=1, k, lower.tail = TRUE, log.p = FALSE) qgengamma.orig(p, shape, scale=1, k, lower.tail = TRUE, log.p = FALSE) rgengamma.orig(n, shape, scale=1, k) Hgengamma.orig(x, shape, scale=1, k) hgengamma.orig(x, shape, scale=1, k) ```

## Arguments

 `x,q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `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 <= x), otherwise, P(X > x).

## Details

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)

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.

## 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.

## Author(s)

Christopher Jackson <[email protected]>

## 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.

`GenGamma`, `GenF.orig`, `GenF`, `Lognormal`, `GammaDist`, `Weibull`.