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.