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\name{Qn}
\alias{Qn}
\alias{Qn.old}
\alias{s_Qn}
%
\title{Robust Location-Free Scale Estimate More Efficient than MAD}
\description{
Compute the robust scale estimator \eqn{Q_n}{Qn}, an efficient
alternative to the MAD.
See the references for more.
}
\usage{
Qn(x, constant = 2.21914, finite.corr = missing(constant))
s_Qn(x, mu.too = FALSE, \dots)
}
\arguments{
\item{x}{numeric vector of observations.}
\item{constant}{number by which the result is multiplied; the default
achieves consistency for normally distributed data. Note that until
Nov. 2010, \dQuote{thanks} to a typo in the very first papers, a slightly
wrong default constant, 2.2219, was used instead of the correct one
which is equal to \code{1 / (sqrt(2) * qnorm(5/8))} (as mentioned
already on p.1277, after (3.7) in Rousseeuw and Croux (1993)).
If you need the old slightly off version for historical
reproducibility, you can use \code{Qn.old()}.
Note that the relative difference is only about 1 in 1000, and that
the correction should not affect the finite sample corrections for
\eqn{n \le 9}{n <= 9}.
}
\item{finite.corr}{logical indicating if the finite sample bias
correction factor should be applied. Defaults to \code{TRUE} unless
\code{constant} is specified.}
\item{mu.too}{logical indicating if the \code{\link[stats]{median}(x)} should
also be returned for \code{s_Qn()}.}
\item{\dots}{potentially further arguments for \code{s_Qn()} passed to
\code{Qn()}.}
}
\value{
\code{Qn()} returns a number, the \eqn{Q_n}{Qn} robust scale
estimator, scaled to be consistent for \eqn{\sigma^2} and
i.i.d. Gaussian observatsions, optionally bias corrected for finite
samples.
\code{s_Qn(x, mu.too=TRUE)} returns a length-2 vector with location
(\eqn{\mu}) and scale; this is typically only useful for
\code{\link{covOGK}(*, sigmamu = s_Qn)}.
}
\details{
As the (default, consistency) constant needed to be corrected,
the finite sample correction has been based on a much more extensive
simulation, and on a 3rd or 4th degree polynomial model in \eqn{1/n}
for odd or even n, respectively.
}
\references{
Rousseeuw, P.J. and Croux, C. (1993)
Alternatives to the Median Absolute Deviation,
\emph{Journal of the American Statistical Association} \bold{88}, 1273--1283.
% MM: ~/save/papers/Rousseeuw/93/R+Croux_MAD_Sn_Qn.pdf
Christophe Croux and Peter J. Rousseeuw (1992)
Time-Efficient Algorithms for Two Highly Robust Estimators of Scale,
\emph{Computational Statistics, Vol. 1}, ed. Dodge and Whittaker,
Physica-Verlag Heidelberg, 411--428; available via Springer Link.
% MM: ~/save/papers/robust-diverse/Croux-Rousseeuw-Timeff_Scale_1992.pdf
%% no longer \url{http://win-www.uia.ac.be/u/statis/abstract/Timeff92.htm}.
About the typo in the \code{constant}:\cr
Christophe Croux (2010)
Private e-mail, Fri Jul 16, w/ Subject
\emph{Re: Slight inaccuracy of Qn implementation \dots\dots}.
}
\seealso{\code{\link[stats]{mad}} for the \sQuote{most robust} but much less efficient
scale estimator; \code{\link{Sn}} for a similar faster but less
efficient alternative. Finally, \code{\link{scaleTau2}} which some
consider \dQuote{uniformly} better than Qn or competitors.
}
\author{Original Fortran code:
Christophe Croux and Peter Rousseeuw \email{rousse@wins.uia.ac.be}.
\cr
Port to C and R: Martin Maechler, \email{maechler@R-project.org}
}
\examples{
set.seed(153)
x <- sort(c(rnorm(80), rt(20, df = 1)))
s_Qn(x, mu.too = TRUE)
Qn(x, finite.corr = FALSE)
}
\keyword{robust}
\keyword{univar}
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