actuar (version 3.3-4)

Loglogistic: 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)

Value

dllogis gives the density,

pllogis gives the distribution function,

qllogis gives the quantile function,

rllogis generates random deviates,

mllogis gives the \(k\)th raw moment, and

levllogis gives the \(k\)th moment of the limited loss variable.

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

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.

Author

Vincent Goulet and Mathieu Pigeon

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 \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), \(-\gamma < k < \gamma\).

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

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)