Description Usage Arguments Details Value Author(s) References See Also

Density, distribution function, quantile function and random generation for the generalized F distribution, using the less flexible original parameterisation described by Prentice (1975).

1 2 3 4 5 6 7 8 9 10 11 | ```
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)
``` |

`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` |
vector of probabilities. |

`n` |
number of observations. If |

If *y ~ F(2*s1, 2*s2)*, and *w =
log(y)**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*, and
*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.

`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.

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

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

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