Density, distribution function, quantile function and random generation for the generalized F distribution, using the less flexible original parameterisation described by Prentice (1975).
Usage
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)
Hgenf.orig(x, mu = 0, sigma = 1, s1, s2)
hgenf.orig(x, mu = 0, sigma = 1, s1, s2)
qgenf.orig(p, mu = 0, sigma = 1, s1, s2, lower.tail = TRUE, log.p = FALSE)
rgenf.orig(n, mu = 0, sigma = 1, s1, s2)Arguments
- x, q
vector of quantiles.
- 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 \(P(X \le x)\), otherwise, \(P(X > x)\).
- p
vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.
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.
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 \(s_1>0,s_2>0\). The probability density function of \(x\) is
$$f(x | \mu, \sigma, s_1, s_2) = \frac{(s_1/s_2)^{s_1} e^{s_1 w}}{\sigma x (1 + s_1 e^w/s_2) ^ {(s_1 + s_2)} B(s_1, s_2)}$$
where \(w = (\log(x) - \mu)/\sigma\), and \(B(s_1,s_2) = \Gamma(s_1)\Gamma(s_2)/\Gamma(s_1+s_2)\) is the beta function.
As \(s_2 \rightarrow \infty\), 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 \(s_1\) 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.
References
R. L. Prentice (1975). Discrimination among some parametric models. Biometrika 62(3):607-614.