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

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     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)

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.

mu

Vector of location parameters.

sigma

Vector of scale parameters.

Q

Vector of first shape parameters.

P

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 <= x), otherwise, P(X > x).

Details

If y ~ F(2*s1, 2*s2), 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 s1,s2.

In this more stable version described by Prentice (1975), s1,s2 are replaced by shape parameters Q,P, with P>0, and

s1 = 2 / (Q^2 + 2P + Q*delta), s2 = 2 / (Q^2 + 2P - Q*delta)

equivalently

Q = (1/s1 - 1/s2) / (1/s1 + 1/s2)^{-1/2}, P = 2 / (s1 + s2)

Define delta = (Q^2 + 2P)^{1/2}, and w = (log(x) - mu)delta / sigma, then the probability density function of x is

f(x) = (delta (s1/s2)^{s1} e^{s1 w}) / (sigma t (1 + s1 e^w/s2) ^ {(s1 + s2)} B(s1, s2))

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.

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.

Note

The parameters Q and P are usually called q and p in the literature - they were made upper-case in these R functions to avoid clashing with the conventional arguments q in the probability function and p in the quantile function.

Author(s)

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

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.

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.

See Also

GenF.orig, GenGamma