llogis: Log-logistic distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Log-logistic distribution

Description

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

Usage

dllogis(x, alpha = 1, beta = 1, log = FALSE)

pllogis(q, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

qllogis(p, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rllogis(n, alpha = 1, beta = 1)

Arguments

x, q	vector of quantiles (x > 0).
alpha	scale parameter (\alpha > 0); equal to the median.
beta	shape parameter (\beta > 0); controls tail heaviness.
log	logical; if TRUE, densities are returned on the log scale.
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].
log.p	logical; if TRUE, probabilities are returned on the log scale.
p	vector of probabilities.
n	number of random values to return.

Details

The log-logistic distribution has density

f(x;\,\alpha,\beta) = \frac{(\beta/\alpha)\,(x/\alpha)^{\beta-1}}{\bigl(1 + (x/\alpha)^\beta\bigr)^2}, \quad x > 0,

where \alpha > 0 is the scale parameter and \beta > 0 is the shape parameter. The scale parameter equals the median. Larger \beta gives lighter tails; the distribution has a finite mean only when \beta > 1 and a finite variance only when \beta > 2.

The log-logistic arises naturally as the distribution of X = e^Y where Y \sim \mathrm{Logistic}(\log\alpha,\, 1/\beta), which yields the numerically convenient log-density

\log f(x) = \log\beta - \log x + \log f_{\mathrm{logistic}}\!\bigl(\beta\log(x/\alpha)\bigr).

The CDF is

F(x;\,\alpha,\beta) = \frac{1}{1 + (x/\alpha)^{-\beta}},

and the quantile function is

Q(p;\,\alpha,\beta) = \alpha \left(\frac{p}{1-p}\right)^{1/\beta}.

dllogis allows for automatic differentiation with RTMB.

Value

dllogis gives the density, pllogis gives the distribution function, qllogis gives the quantile function, and rllogis generates random deviates.

Examples

x <- rllogis(5, alpha = 1, beta = 2)
dllogis(x, alpha = 1, beta = 2)
pllogis(c(0.5, 1, 2), alpha = 1, beta = 2)
qllogis(c(0.25, 0.5, 0.75), alpha = 1, beta = 2)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.