Generalized F distribution (original parameterisation)
Description
Density, distribution function, quantile function and random generation for the generalized F distribution, using the less flexible original parameterisation described by Prentice (1975).
Usage
1 2 3 4 5 6  dgenf.orig(x, mu=0, sigma=1, s1, s2, log = FALSE)
pgenf.orig(q, mu=0, sigma=1, s1, s2, lower.tail = TRUE, log.p = FALSE)
qgenf.orig(p, mu=0, sigma=1, s1, s2, lower.tail = TRUE, log.p = FALSE)
rgenf.orig(n, mu=0, sigma=1, s1, s2)
Hgenf.orig(x, mu=0, sigma=1, s1, s2)
hgenf.orig(x, mu=0, sigma=1, s1, s2)

Arguments
x,q 
vector of quantiles. 
p 
vector of probabilities. 
n 
number of observations. If 
mu 
Vector of location parameters. 
sigma 
Vector of scale parameters. 
s1 
Vector of first F shape parameters. 
s2 
vector of second F 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>0,s2>0. The probability density function of x is
f(x  mu, sigma, s_1, s_2) = ((s1/s2)^{s1} e^{s1 w}) / (sigma x (1 + s1 e^w/s2) ^ (s1 + s2) B(s1, s2))
where w = (log(x)  mu)/sigma , B(s1,s2) = Γ(s1)Γ(s2)/Γ(s1+s2) is the beta function.
As s2 > infinity, the distribution of x tends towards an original generalized gamma distribution with the following parameters:
dgengamma.orig(x, shape=1/sigma, scale=exp(mu) / s1^sigma, k=s1)
See GenGamma.orig
for how this includes several other
common distributions as special cases.
The alternative parameterisation of the generalized F distribution,
originating from Prentice (1975) and given in this package as
GenF
, is
preferred for statistical modelling, since it is more stable as
s1 tends to infinity, and includes a further new class
of distributions with negative first shape parameter. The original
is provided here for the sake of completion and compatibility.
Value
dgenf.orig
gives the density, pgenf.orig
gives the distribution
function, qgenf.orig
gives the quantile function, rgenf.orig
generates random deviates, Hgenf.orig
retuns the cumulative hazard
and hgenf.orig
the hazard.
Author(s)
Christopher Jackson <chris.jackson@mrcbsu.cam.ac.uk>
References
R. L. Prentice (1975). Discrimination among some parametric models. Biometrika 62(3):607614.
See Also
GenF
, GenGamma.orig
, GenGamma