Density, distribution function, quantile function and random generation for the log-logistic distribution with shape and scale parameters.
Usage
dloglogistic(x, shape, scale, log = FALSE)
ploglogistic(q, shape, scale, lower.tail = TRUE, log.p = FALSE)
qloglogistic(p, shape, scale, lower.tail = TRUE, log.p = FALSE)
rloglogistic(n, shape, scale)Arguments
- x, q
vector of quantiles.
- shape
shape parameter. Must be positive.
- scale
scale parameter. Must be positive.
- log, log.p
logical; if
TRUE, probabilities/densities p are returned 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.
Value
dloglogistic gives the density, ploglogistic gives the distribution
function, qloglogistic gives the quantile function, and rloglogistic
generates random deviates.
The length of the result is determined by n for rloglogistic, and is the
maximum of the lengths of the numerical arguments for the other functions.
Details
The log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is the probability distribution of a random variable whose logarithm has a logistic distribution.
The probability density function is given by: $$f(x) = \frac{(\alpha/\gamma)(x/\gamma)^{\alpha-1}}{(1+(x/\gamma)^\alpha)^2}$$ for \(x \geq 0\), where \(\alpha\) is the shape parameter and \(\gamma\) is the scale parameter.
The cumulative distribution function is: $$F(x) = \frac{1}{1+(x/\gamma)^{-\alpha}}$$
Examples
# Density at x = 1 with shape = 2 and scale = 1
dloglogistic(1, shape = 2, scale = 1)
#> [1] 0.5
# CDF at x = 1
ploglogistic(1, shape = 2, scale = 1)
#> [1] 0.5
# Quantile for p = 0.5 (median)
qloglogistic(0.5, shape = 2, scale = 1)
#> [1] 1
# Generate 10 random values
rloglogistic(10, shape = 2, scale = 1)
#> [1] 7.0983521 1.6937515 0.2328874 1.0623661 1.5124712 1.4868928 0.1795469
#> [8] 0.5396849 0.6559489 1.3231672