Density, distribution function, hazards, quantile function and random generation for the log-logistic distribution.

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.

shape, scale

vector of shape and scale 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, \(P(X > x)\).

`dllogis`

gives the density, `pllogis`

gives the
distribution function, `qllogis`

gives the quantile function,
`hllogis`

gives the hazard function, `Hllogis`

gives the
cumulative hazard function, and `rllogis`

generates random
deviates.

The log-logistic distribution with `shape`

parameter
\(a>0\) and `scale`

parameter \(b>0\) has probability
density function

$$f(x | a, b) = (a/b) (x/b)^{a-1} / (1 + (x/b)^a)^2$$

and hazard

$$h(x | a, b) = (a/b) (x/b)^{a-1} / (1 + (x/b)^a)$$

for \(x>0\). The hazard is decreasing for shape \(a\leq 1\), and unimodal for \(a > 1\).

The probability distribution function is $$F(x | a, b) = 1 - 1 / (1 + (x/b)^a)$$

If \(a > 1\), the mean is \(b c / sin(c)\), and if \(a > 2\) the variance is \(b^2 * (2*c/sin(2*c) - c^2/sin(c)^2)\), where \(c = \pi/a\), otherwise these are undefined.

Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.