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\name{mc}
\alias{mc}
\title{Medcouple, a Robust Measure of Skewness}
\description{
Compute the \sQuote{medcouple}, a \emph{robust} concept and estimator
of skewness. The medcouple is defined as a scaled median difference
of the left and right half of distribution, and hence \emph{not} based
on the third moment as the classical skewness.
}
\usage{
mc(x, na.rm = FALSE, doReflect = (length(x) <= 100),
doScale = TRUE, # <- chg default to 'FALSE' ?
eps1 = 1e-14, eps2 = 1e-15, # << new in 0.93-2 (2018-07..)
maxit = 100, trace.lev = 0, full.result = FALSE)
}
\arguments{
\item{x}{a numeric vector}
\item{na.rm}{logical indicating how missing values (\code{\link{NA}}s)
should be dealt with.}
\item{doReflect}{logical indicating if the internal MC should also be
computed on the \emph{reflected} sample \code{-x}, with final result
\code{(mc.(x) - mc.(-x))/2}. This makes sense since the internal
MC, \code{mc.()} computes the himedian() which can differ slightly from
the median.}%% only whenever sum(x <= med) * sum(x >= med) is even
\item{doScale}{logical indicating if the internal algorithm should
also \emph{scale} the data (using the most distant value from the
median which is unrobust and numerically dangerous); scaling has been
the hardwired default in the original algorithm and in \R's
\code{mc()} till summer 2018.}
\item{eps1, eps2}{tolerance in the algorithm; only change with care!}
\item{maxit}{maximul number of iterations; typically a few should be
sufficient.}
\item{trace.lev}{integer specifying how much diagnostic output the
algorithm (in C) should produce. No output by default, most output
for \code{trace.lev = 5}.}
\item{full.result}{logical indicating if the full return values (from
C) should be returned as a list via \code{attr(*, "mcComp")}.}
}
% \details{
% ~~ If necessary, more details than the description above ~~
% }
\value{
a number between -1 and 1, which is the medcouple, \eqn{MC(x)}.
For \code{r <- mc(x, full.result = TRUE, ....)}, then
\code{attr(r, "mcComp")} is a list with components
\item{medc}{the medcouple \eqn{mc.(x)}.}
\item{medc2}{the medcouple \eqn{mc.(-x)} if \code{doReflect=TRUE}.}
\item{eps}{tolerances used.}
\item{iter,iter2}{number of iterations used.}
\item{converged,converged2}{logical specifying \dQuote{convergence}.}
}
\section{Convergence Problems}{
For extreme cases there \emph{are} convergence problems.
Some of them can be alleviated by \dQuote{loosening} the tolerances
\code{eps1} and \code{eps2}.
\cr
For others, with pecular values, notably many almost-ties with the
median, it can help extremely to replace \code{mc(x, *)} by
\code{mc(jitter(x), *)}.% MM: see also ~/R/MM/Pkg-ex/robustbase/Robnik-mc.R
Also, the algorithm not only centers the data around the median but
also scales them by the extremes which may have a negative effect
e.g., when changing an extreme outlier to even more extreme, the
result changes wrongly; see the 'mc10x' example.
}
\references{
Guy Brys, Mia Hubert and Anja Struyf (2004)
A Robust Measure of Skewness;
\emph{JCGS} \bold{13} (4), 996--1017.
Hubert, M. and Vandervieren, E. (2008).
An adjusted boxplot for skewed distributions,
\emph{Computational Statistics and Data Analysis} \bold{52}, 5186--5201.
}
\author{Guy Brys; modifications by Tobias Verbeke and bug fixes and
extensions by Manuel Koller and Martin Maechler.
}
\seealso{\code{\link{Qn}} for a robust measure of scale (aka
\dQuote{dispersion}), ....
}
\examples{
mc(1:5) # 0 for a symmetric sample
x1 <- c(1, 2, 7, 9, 10)
mc(x1) # = -1/3
data(cushny)
mc(cushny) # 0.125
stopifnot(mc(c(-20, -5, -2:2, 5, 20)) == 0,
mc(x1, doReflect=FALSE) == -mc(-x1, doReflect=FALSE),
all.equal(mc(x1, doReflect=FALSE), -1/3, tolerance = 1e-12))
## Susceptibility of the current algorithm to large outliers :
dX10 <- function(X) c(1:5,7,10,15,25, X) # generate skewed size-10 with 'X'
x <- c(10,20,30, 100^(1:20))
(mc10x <- vapply(x, function(X) mc(dX10(X)), 1))
## limit X -> Inf should be 7/12 = 0.58333... but that "breaks down a bit" :
plot(x, mc10x, type="b", main = "mc( c(1:5,7,10,15,25, X) )", xlab="X", log="x")
}
\keyword{robust}
\keyword{univar}
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