Generalized gamma distribution (original parameterisation)
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 
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
lognormal, 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 <chris.jackson@mrcbsu.cam.ac.uk>
References
Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:118792.
Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539544.
Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409419.
See Also
GenGamma
, GenF.orig
, GenF
,
Lognormal
, GammaDist
, Weibull
.