dgenf(x, mu=0, sigma=1, Q, P, log = FALSE) pgenf(q, mu=0, sigma=1, Q, P, lower.tail = TRUE, log.p = FALSE) qgenf(p, mu=0, sigma=1, Q, P, lower.tail = TRUE, log.p = FALSE) rgenf(n, mu=0, sigma=1, Q, P) Hgenf(x, mu=0, sigma=1, Q, P) hgenf(x, mu=0, sigma=1, Q, P)length(n) > 1, the length is
taken to be the number required.dgenf gives the density, pgenf gives the distribution
function, qgenf gives the quantile function, rgenf
generates random deviates, Hgenf retuns the cumulative hazard
and hgenf the hazard.
In this more stable version described by Prentice (1975), $s1,s2$ are replaced by shape parameters $Q,P$, with $P>0$, and
$$s_1 = 2(Q^2 + 2P + Q\delta)^{-1}, \quad s_2 = 2(Q^2 + 2P - Q\delta)^{-1}$$ equivalently $$Q = \left(\frac{1}{s_1} - \frac{1}{s_2}\right)\left(\frac{1}{s_1} + \frac{1}{s_2}\right)^{-1/2}, \quad P = \frac{2}{s_1 + s_2} $$
Define $delta = (Q^2 + 2P)^{1/2}$,
and $w = (log(x) - mu)delta /
sigma$, then the probability density function of $x$ is $$ f(x) = \frac{\delta (s_1/s_2)^{s_1} e^{s_1 w}}{\sigma x (1 + s_1 e^w/s_2) ^ {(s_1 + s_2)} B(s_1, s_2)}
$$
The original parameterisation is available in this package as
dgenf.orig, for the sake of completion / compatibility.
With the above definitions,
dgenf(x, mu=mu, sigma=sigma, Q=Q, P=P) = dgenf.orig(x, mu=mu, sigma=sigma/delta, s1=s1, s2=s2)
The generalized F distribution with P=0 is equivalent to the
generalized gamma distribution dgengamma, so that
dgenf(x, mu, sigma, Q, P=0) equals dgengamma(x, mu, sigma, Q).
The generalized gamma reduces further to several
common distributions, as described in the GenGamma
help page.
The generalized F distribution includes the log-logistic distribution
(see Llogis) as a further special case:
dgenf(x, mu=mu, sigma=sigma, Q=0, P=1) = dllogis(x, shape=sqrt(2)/sigma, scale=exp(mu))
The range of hazard trajectories available under this distribution are discussed in detail by Cox (2008). Jackson et al. (2010) give an application to modelling oral cancer survival for use in a health economic evaluation of screening.
Cox, C. (2008). The generalized $F$ distribution: An umbrella for parametric survival analysis. Statistics in Medicine 27:4301-4312.
Jackson, C. H. and Sharples, L. D. and Thompson, S. G. (2010). Survival models in health economic evaluations: balancing fit and parsimony to improve prediction. International Journal of Biostatistics 6(1):Article 34.
GenF.orig, GenGamma