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 distributiondgamma. 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