Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the original parameterisation from Stacy (1962).
Usage
dgengamma.orig(x, shape, scale = 1, k, log = FALSE)
pgengamma.orig(q, shape, scale = 1, k, lower.tail = TRUE, log.p = FALSE)
Hgengamma.orig(x, shape, scale = 1, k)
hgengamma.orig(x, shape, scale = 1, k)
qgengamma.orig(p, shape, scale = 1, k, lower.tail = TRUE, log.p = FALSE)
rgengamma.orig(n, shape, scale = 1, k)Arguments
- x, q
vector of quantiles.
- 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 \(P(X \le x)\), 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.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.
Details
If \(w \sim Gamma(k,1)\), 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
$$f(x | a, b, k) = \frac{b}{\Gamma(k)} \frac{x^{bk - 1}}{a^{bk}} $$$$ \exp(-(x/a)^b)$$
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.
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.