Log-logistic distribution
llogis.RdDensity, 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.
Value
dllogis gives the density, pllogis gives the distribution function,
qllogis gives the quantile function, and rllogis generates random deviates.
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.