# 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.