Llogis: The log-logistic distribution

Description

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

Usage

dllogis(x, shape = 1, scale = 1, log = FALSE)
pllogis(q, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
qllogis(p, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
rllogis(n, shape = 1, scale = 1)
hllogis(x, shape = 1, scale = 1, log = FALSE)
Hllogis(x, shape = 1, scale = 1, log = FALSE)

Value

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.

Arguments

x, q: vector of quantiles.
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, $P(X > x)$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Author

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

Details

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.

References

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