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% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/stable.r
\name{stable.mode}
\alias{stable.mode}
\title{Mode of a Stable Distribution}
\usage{
stable.mode(loc, disp, skew, tail)
}
\arguments{
\item{loc}{vector of (real) location parameters.}
\item{disp}{vector of (positive) dispersion parameters.}
\item{skew}{vector of skewness parameters (in [-1,1]).}
\item{tail}{vector of parameters (in [1,2]) related to the tail thickness.}
}
\value{
A list of size 3 giving the mode, \eqn{a} and \eqn{b}.
}
\description{
This function gives a reliable approximation to the mode of a stable
distribution with location, dispersion, skewness and tail thickness
specified by the parameters \code{loc}, \code{disp}, \code{skew} and
\code{tail}. \code{tail} must be in (1,2).
}
\details{
\code{loc} is a location parameter in the same way as the mean in the normal
distribution: it can take any real value.
\code{disp} is a dispersion parameter in the same way as the standard
deviation in the normal distribution: it can take any positive value.
\code{skew} is a skewness parameter: it can take any value in \eqn{(-1,1)}.
The distribution is right-skewed, symmetric and left-skewed when \code{skew}
is negative, null or positive respectively.
\code{tail} is a tail parameter (often named the characteristic exponent):
it can take any value in \eqn{(0,2)} (with \code{tail=1} and \code{tail=2}
yielding the Cauchy and the normal distributions respectively when symmetry
holds).
The simplest empirical formula found to give a satisfactory approximation to
the mode for values of \code{tail} in \eqn{(1,2)} is \deqn{
loc+disp*a*skew*exp(-b*abs(skew))} with \deqn{ a =
1.7665114+1.8417675*tail-2.2954390*tail^2+0.4666749*tail^3} and \deqn{ b =
-0.003142967+632.4715*tail*exp(-7.106035*tail)}.
}
\examples{
x <- seq(-5,5,by=0.1)
plot(x,dstable(x,loc=0,disp=1,skew=-1,tail=1.5),type="l",ylab="f(x)")
xhat <- stable.mode(loc=0,disp=1,skew=-1,tail=1.5)$ytilde
fxhat <- dstable(xhat,loc=0,disp=1,skew=-1,tail=1.5)
lines(c(xhat,xhat),c(0,fxhat),lty="dotted")
}
\references{
Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns
using regression models based on non-symmetric stable distributions. Applied
Statistics, 48, 409-424.
}
\seealso{
\code{stable} for more details on the stable
distribution.
\code{stablereg} to fit generalized linear models for the
stable distribution parameters.
}
\author{
Philippe Lambert (Catholic University of Louvain, Belgium,
\email{phlambert@stat.ucl.ac.be}) and Jim Lindsey.
}
\keyword{distribution}
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