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\name{Loglogistic}
\alias{Loglogistic}
\alias{dllogis}
\alias{pllogis}
\alias{qllogis}
\alias{rllogis}
\alias{mllogis}
\alias{levllogis}
\title{The Loglogistic Distribution}
\description{
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Loglogistic distribution with
parameters \code{shape} and \code{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{
\item{x, q}{vector of quantiles.}
\item{p}{vector of probabilities.}
\item{n}{number of observations. If \code{length(n) > 1}, the length is
taken to be the number required.}
\item{shape, scale}{parameters. Must be strictly positive.}
\item{rate}{an alternative way to specify the scale.}
\item{log, log.p}{logical; if \code{TRUE}, probabilities/densities
\eqn{p} are returned as \eqn{\log(p)}{log(p)}.}
\item{lower.tail}{logical; if \code{TRUE} (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}.}
\item{order}{order of the moment.}
\item{limit}{limit of the loss variable.}
}
\details{
The loglogistic distribution with parameters \code{shape} \eqn{=
\gamma}{= a} and \code{scale} \eqn{= \theta}{= s} has density:
\deqn{f(x) = \frac{\gamma (x/\theta)^\gamma}{%
x [1 + (x/\theta)^\gamma]^2}}{%
f(x) = a (x/s)^a / (x [1 + (x/s)^a]^2)}
for \eqn{x > 0}, \eqn{\gamma > 0}{a > 0} and \eqn{\theta > 0}{b > 0}.
The \eqn{k}th raw moment of the random variable \eqn{X} is
\eqn{E[X^k]}, \eqn{-\gamma < k < \gamma}{-shape < k < shape}.
The \eqn{k}th limited moment at some limit \eqn{d} is \eqn{E[\min(X,
d)^k]}{E[min(X, d)^k]}, \eqn{k > -\gamma}{k > -shape}
and \eqn{1 - k/\gamma}{1 - k/shape} not a negative integer.
}
\value{
\code{dllogis} gives the density,
\code{pllogis} gives the distribution function,
\code{qllogis} gives the quantile function,
\code{rllogis} generates random deviates,
\code{mllogis} gives the \eqn{k}th raw moment, and
\code{levllogis} gives the \eqn{k}th moment of the limited loss
variable.
Invalid arguments will result in return value \code{NaN}, with a warning.
}
\note{
\code{levllogis} computes the limited expected value using
\code{\link{betaint}}.
Also known as the Fisk distribution. See also Kleiber and Kotz (2003)
for alternative names and parametrizations.
The \code{"distributions"} package vignette provides the
interrelations between the continuous size distributions in
\pkg{actuar} and the complete formulas underlying the above functions.
}
\references{
Kleiber, C. and Kotz, S. (2003), \emph{Statistical Size Distributions
in Economics and Actuarial Sciences}, Wiley.
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012),
\emph{Loss Models, From Data to Decisions, Fourth Edition}, Wiley.
}
\seealso{
\code{\link{dpareto3}} for an equivalent distribution with a location
parameter.
}
\author{
Vincent Goulet \email{vincent.goulet@act.ulaval.ca} and
Mathieu Pigeon
}
\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)
}
\keyword{distribution}
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