Generalized Gamma Probability Distribution

Description

Fast implementation of density, distribution function, quantile function and random generation for the Generalized Gamma probability distribution.

Usage

dggamma(x, a, b, k, log = F)

pggamma(q, a, b, k, lower.tail = TRUE, log.p = FALSE)

qggamma(p, a, b, k, lower.tail = TRUE, log.p = FALSE)

rggamma(n, a, b, k)

Arguments

Details

The generalized gamma distribution proposed by Stacy (1962) has parameters a, d, p, but here we adopt the reparametrization

a = a

b = p

k = \frac{d}{p}

as is used by the R package *flexsurv*.

Probability density function

f(x) = \frac{b x^{bk-1} \exp[-(x/a)^b]}{a^{bk} \Gamma(k)}

Cumulative density function

F(x) = \frac{\gamma(k, (x/a)^b)}{\Gamma(k)}

The above function can be written in terms of a Gamma(\alpha, \beta). Let T \sim Gamma(k, 1) and its cumulative distribution be denoted as F_T(t), then the cumulative density function of the generalized gamma distribution can be written as

F(x) = F_T( (x/a)^b )

which allows us to write the quantile function of the generalized gamma in terms of the gamma one (Q_T(u) is the quantile function of T)

Q(u) = (Q_T(u) \cdot a)^{1/b}

from which random numbers can be drawn.

References

Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of mathematical statistics, 33(3), 1187-1192.

Examples

x = seq(0.001, 5, length=1000);
plot(x, dggamma(x, 3, 1.8, 0.5), col=2, type="l", lwd=4, ylim=c(0, 1));
lines(x, pggamma(x, 3, 1.8, 0.5), col=4, type="l", lwd=4, ylim=c(0, 1));
legend("right", c("PDF", "CDF"), col=c(2, 4), lwd=4);

r = rgamma(n = 100, 2, 2);
lik = function(params) -sum(dggamma(r, params[1], params[2], params[3], log=TRUE));
optPar = optim(lik, par=c(1, 1, 1), method="L-BFGS", lower=0.00001, upper=Inf)$par;
x = seq(0.001, 5, length=1000);
plot(x, dgamma(x, 2, 2), type="l", col=2, lwd=4, ylim=c(0, 1));
lines(x, dggamma(x, optPar[1], optPar[2], optPar[3]), col=4, lwd=4);
legend("topright", c("Gamma(shape=2, rate=2)", "MLE Gen. Gamma"), col=c(2, 4), lwd=4);

Generalized Gamma Probability Distribution

Description

Density, distribution function, quantile function and random generation for the Generalized Gamma lifetime distributions.

Details

This package follows naming convention that is consistent with base R, where density (or probability mass) functions, distribution functions, quantile functions and random generation functions names are followed by d, p, q, and r prefixes.

Behaviour of the functions is consistent with base R, where for not valid parameters values NaN's are returned, while for values beyond function support 0's are returned (e.g. for non-integers in discrete distributions, or for negative values in functions with non-negative support).

C++ was not used, as the R code proved itself most efficient. See the package website page for more details.