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

Details

The Loglogistic distribution with parameters shape = a and scale = s has density:

f(x) = a (x/s)^a / (x [1 + (x/s)^a]^2)

for x > 0, a > 0 and b > 0.

The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)^k].

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

Also known as the Fisk distribution.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2008), Loss Models, From Data to Decisions, Third Edition, Wiley.

Examples

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

Results