\name{shorth}
\alias{shorth}
\title{A location estimator based on the shorth}
\description{A location estimator based on the shorth}
\usage{shorth(x, na.rm=FALSE, tie.action="mean", tie.limit=0.05)}
\arguments{
  \item{x}{Numeric}
  \item{na.rm}{Logical. If \code{TRUE}, then non-finite (according to
    \code{\link{is.finite}}) values in \code{x} are ignored. Otherwise,
    presence of non-finite or \code{NA} values will lead to an error message.}
  \item{tie.action}{Character scalar. See details.}
  \item{tie.limit}{Numeric scalar. See details.}
}

\details{The shorth is the shortest interval that covers half of the
  values in \code{x}. This function calculates the mean of the \code{x}
  values that lie in the shorth. This was proposed by Andrews (1972) as a 
  robust estimator of location.

  Ties: if there are multiple shortest intervals,
  the action specified in \code{ties.action} is applied.
  Allowed values are \code{mean} (the default), \code{max} and \code{min}.
  For \code{mean}, the average value is considered; however, an error is
  generated if the start indices of the different shortest intervals
  differ by more than the fraction \code{tie.limit} of \code{length(x)}.
  For \code{min} and \code{max}, the left-most or right-most, respectively, of
  the multiple shortest intervals is considered.

  Rate of convergence: as an estimator of location of a unimodal
  distribution, under regularity conditions,
  the quantity computed here has an asymptotic rate of only $n^{-1/3}$ and a
  complicated limiting distribution. 

  See \code{\link{half.range.mode}} for an iterative version
  that refines the estimate iteratively and has a builtin bootstrapping option.
}

\value{The mean of the \code{x} values that lie in the shorth.}

\references{
  \itemize{
    \item G Sawitzki, \dQuote{The Shorth Plot.}
    Available at http://lshorth.r-forge.r-project.org/TheShorthPlot.pdf 

    \item DF Andrews, \dQuote{Robust Estimates of Location.}
    Princeton University Press (1972).

    \item R Grueble, \dQuote{The Length of the Shorth.} Annals of
    Statistics 16, 2:619-628 (1988).

    \item DR Bickel and R Fruehwirth, \dQuote{On a fast, robust
    estimator of the mode: Comparisons to other robust estimators
    with applications.} Computational Statistics & Data Analysis
    50, 3500-3530 (2006).
  }
}

\author{Wolfgang Huber \url{http://www.ebi.ac.uk/huber}, Ligia Pedroso Bras}
\seealso{\code{\link{half.range.mode}}}

\examples{
 
  x = c(rnorm(500), runif(500) * 10)
  methods = c("mean", "median", "shorth", "half.range.mode")
  ests = sapply(methods, function(m) get(m)(x))

  if(interactive()) {
    colors = 1:4
    hist(x, 40, col="orange")
    abline(v=ests, col=colors, lwd=3, lty=1:2)
    legend(5, 100, names(ests), col=colors, lwd=3, lty=1:2) 
  }
}
\keyword{arith}