# GenF.orig: Generalized F distribution (original parameterisation) In flexsurv: Flexible parametric survival models

## 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 `length(n) > 1`, the length is taken to be the number required. `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 <[email protected]>

## References

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

`GenF`, `GenGamma.orig`, `GenGamma`