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Loglogistic: The Loglogistic Distribution

Description

Density, 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.

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.

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 $E[X^k]$ and the $k$ limited moment at some limit $d$ is $E[\min(X, d)]$.

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

Run this code
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)

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