Loglogistic: The Loglogistic Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Loglogistic Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Loglogistic distribution with parameters shape and scale.

Usage

dllogis(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pllogis(q, shape, rate = 1, scale = 1/rate,
        lower.tail = TRUE, log.p = FALSE)
qllogis(p, shape, rate = 1, scale = 1/rate,
        lower.tail = TRUE, log.p = FALSE)
rllogis(n, shape, rate = 1, scale = 1/rate)
mllogis(order, shape, rate = 1, scale = 1/rate)
levllogis(limit, shape, rate = 1, scale = 1/rate,
          order = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
shape, scale	parameters. Must be strictly positive.
rate	an alternative way to specify the scale.
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].
order	order of the moment.
limit	limit of the loss variable.

Details

The loglogistic distribution with parameters shape = \gamma and scale = \theta has density:

f(x) = \frac{\gamma (x/\theta)^\gamma}{% x [1 + (x/\theta)^\gamma]^2}

for x > 0, \gamma > 0 and \theta > 0.

The kth raw moment of the random variable X is E[X^k], -\gamma < k < \gamma.

The kth limited moment at some limit d is E[\min(X, d)^k], k > -\gamma and 1 - k/\gamma not a negative integer.

Value

dllogis gives the density, pllogis gives the distribution function, qllogis gives the quantile function, rllogis generates random deviates, mllogis gives the kth raw moment, and levllogis gives the kth moment of the limited loss variable.

Invalid arguments will result in return value NaN, with a warning.

Note

levllogis computes the limited expected value using betaint.

Also known as the Fisk distribution. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

The "distributions" package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

Author(s)

Vincent Goulet [e-mail] and Mathieu Pigeon

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

dpareto3 for an equivalent distribution with a location parameter.

Examples

exp(dllogis(2, 3, 4, log = TRUE))
p <- (1:10)/10
pllogis(qllogis(p, 2, 3), 2, 3)

## mean
mllogis(1, 2, 3)

## case with 1 - order/shape > 0
levllogis(10, 2, 3, order = 1)

## case with 1 - order/shape < 0
levllogis(10, 2/3, 3, order = 1)

actuar documentation built on Aug. 9, 2025, 1:06 a.m.