GenF: Generalized F distribution

Description

Density, distribution function, hazards, quantile function and random generation for the generalized F distribution, using the reparameterisation by Prentice (1975).

Usage

dgenf(x, mu = 0, sigma = 1, Q, P, log = FALSE)
pgenf(q, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE)
Hgenf(x, mu = 0, sigma = 1, Q, P)
hgenf(x, mu = 0, sigma = 1, Q, P)
qgenf(p, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE)
rgenf(n, mu = 0, sigma = 1, Q, P)

Arguments

x, q

Vector of quantiles.

Vector of location parameters.

sigma

Vector of scale parameters.

Vector of first shape parameters.

Vector of second 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, $P(X > x)$.

Vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required.

Value

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.

Details

If $y \sim F(2s_1, 2s_2)$, and $w = $$ \log(y)$ then $x = \exp(w\sigma + \mu)$ has the original generalized F distribution with location parameter $\mu$, scale parameter $\sigma>0$ and shape parameters $s_1,s_2$.

In this more stable version described by Prentice (1975), $s_1,s_2$ 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.

References

R. L. Prentice (1975). Discrimination among some parametric models. Biometrika 62(3):607-614.

Cox, C. (2008). The generalized $F$ distribution: An umbrella for parametric survival analysis. Statistics in Medicine 27:4301-4312.