Density, distribution function, hazards, quantile function and random generation for the generalized F distribution, using the reparameterisation by Prentice (1975).

## Usage

```
dgenf(x, mu = 0, sigma = 1, Q, P, log = FALSE)
pgenf(q, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE)
Hgenf(x, mu = 0, sigma = 1, Q, P)
hgenf(x, mu = 0, sigma = 1, Q, P)
qgenf(p, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE)
rgenf(n, mu = 0, sigma = 1, Q, P)
```

## Arguments

- x, q
Vector of quantiles.

- mu
Vector of location parameters.

- sigma
Vector of scale parameters.

- Q
Vector of first shape parameters.

- P
Vector of second 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`

gives the density, `pgenf`

gives the distribution
function, `qgenf`

gives the quantile function, `rgenf`

generates
random deviates, `Hgenf`

retuns the cumulative hazard and `hgenf`

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 \(s_1,s_2\).

In this more stable version described by Prentice (1975), \(s_1,s_2\) are replaced by shape parameters \(Q,P\), with \(P>0\), and

$$s_1 = 2(Q^2 + 2P + Q\delta)^{-1}, \quad s_2 = 2(Q^2 + 2P - Q\delta)^{-1}$$ equivalently $$Q = \left(\frac{1}{s_1} - \frac{1}{s_2}\right)\left(\frac{1}{s_1} + \frac{1}{s_2}\right)^{-1/2}, \quad P = \frac{2}{s_1 + s_2} $$

Define \(\delta = (Q^2 + 2P)^{1/2}\), and
\(w = (\log(x) - \mu)\delta /\sigma\),
then the probability density function of \(x\) is $$ f(x) =
\frac{\delta (s_1/s_2)^{s_1} e^{s_1 w}}{\sigma x (1 + s_1 e^w/s_2) ^ {(s_1
+ s_2)} B(s_1, s_2)} $$ The
original parameterisation is available in this package as
`dgenf.orig`

, for the sake of completion / compatibility. With
the above definitions,

```
dgenf(x, mu=mu, sigma=sigma, Q=Q, P=P) = dgenf.orig(x, mu=mu,
sigma=sigma/delta, s1=s1, s2=s2)
```

The generalized F distribution with `P=0`

is equivalent to the
generalized gamma distribution `dgengamma`

, so that
`dgenf(x, mu, sigma, Q, P=0)`

equals ```
dgengamma(x, mu, sigma,
Q)
```

. The generalized gamma reduces further to several common
distributions, as described in the `GenGamma`

help page.

The generalized F distribution includes the log-logistic distribution (see
`eha::dllogis`

) as a further special case:

```
dgenf(x, mu=mu, sigma=sigma, Q=0, P=1) = eha::dllogis(x,
shape=sqrt(2)/sigma, scale=exp(mu))
```

The range of hazard trajectories available under this distribution are discussed in detail by Cox (2008). Jackson et al. (2010) give an application to modelling oral cancer survival for use in a health economic evaluation of screening.

## Note

The parameters `Q`

and `P`

are usually called \(q\) and
\(p\) in the literature - they were made upper-case in these R functions
to avoid clashing with the conventional arguments `q`

in the
probability function and `p`

in the quantile function.

## References

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

Cox, C. (2008). The generalized \(F\) distribution: An umbrella for parametric survival analysis. Statistics in Medicine 27:4301-4312.

Jackson, C. H. and Sharples, L. D. and Thompson, S. G. (2010). Survival models in health economic evaluations: balancing fit and parsimony to improve prediction. International Journal of Biostatistics 6(1):Article 34.