Density, distribution function, hazards, quantile function and random generation for the log-logistic distribution.
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)
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.
vector of quantiles.
vector of shape and scale parameters.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P(X \le x)\), otherwise, \(P(X > x)\).
vector of probabilities.
number of observations. If length(n) > 1
, the
length is taken to be the number required.
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
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.