Generalized gamma distribution (original parameterisation)

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Description

Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the original parameterisation from Stacy (1962).

Usage

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     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 <chris.jackson@mrc-bsu.cam.ac.uk>

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.

See Also

GenGamma, GenF.orig, GenF, Lognormal, GammaDist, Weibull.