GenGamma  R Documentation 
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) loggamma distribution.
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)
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
(logscale) standard deviation of the lognormal distribution, “sdlog” in

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 <= x), otherwise, P(X > x). 
p 
vector of probabilities. 
n 
number of observations. If 
If g ~ Gamma(Q^{2}, 1) , and w = log(Q^2*g) / Q, then x = exp(mu + sigma w) follows the generalized gamma distribution with probability density function
f(x  mu, sigma, Q) = Q (Q^{2})^{Q^{2}} / (sigma * x * Gamma(Q^{2})) exp(Q^{2}*(Q*w  exp(Q*w)))
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 lognormal
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, lognormal 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.
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.
Christopher Jackson <chris.jackson@mrcbsu.cam.ac.uk>
Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539544.
Farewell, V. T. and Prentice, R. L. (1977). A study of distributional shape in life testing. Technometrics 19(1):6975.
Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409419.
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 26:42524374
Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:118792
GenGamma.orig
, GenF
,
Lognormal
, GammaDist
, Weibull
.
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