flexsurv (version 1.1.1)

# GenGamma: Generalized gamma distribution

## Description

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 distribution dgamma. 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 Mu<U+00F1>oz, 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

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