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% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/lientz.R
\name{lientz}
\alias{lientz}
\alias{Lientz}
\alias{plot.lientz}
\alias{print.lientz}
\alias{mlv.lientz}
\title{The empirical Lientz function and the Lientz mode estimator}
\usage{
lientz(x, bw = NULL)
\method{plot}{lientz}(x, zoom = FALSE, ...)
\method{print}{lientz}(x, digits = NULL, ...)
\method{mlv}{lientz}(x, bw = NULL, abc = FALSE, par = shorth(x), optim.method = "BFGS", ...)
}
\arguments{
\item{x}{numeric (vector of observations) or an object of class \code{"lientz"}.}
\item{bw}{numeric. The smoothing bandwidth to be used.
Should belong to (0, 1). Parameter 'beta' in Lientz (1970) function.}
\item{zoom}{logical. If \code{TRUE}, one can zoom on the graph created.}
\item{...}{if \code{abc = FALSE}, further arguments to be passed to
\code{\link[stats]{optim}}, or further arguments to be passed to
\code{\link[graphics]{plot}}.}
\item{digits}{numeric. Number of digits to be printed.}
\item{abc}{logical. If \code{FALSE} (the default), the Lientz empirical function
is minimised using \code{\link[stats]{optim}}.}
\item{par}{numeric. The initial value used in \code{\link[stats]{optim}}.}
\item{optim.method}{character. If \code{abc = FALSE}, the method used in
\code{\link[stats]{optim}}.}
}
\value{
\code{lientz} returns an object of class \code{c("lientz", "function")};
this is a function with additional attributes:
\itemize{
\item{x}{ the \code{x} argument}
\item{bw}{ the \code{bw} argument }
\item{call}{ the call which produced the result }
}
\code{mlv.lientz} returns a numeric value, the mode estimate.
If \code{abc = TRUE}, the \code{x} value minimizing the Lientz empirical
function is returned. Otherwise, the \code{\link[stats]{optim}} method is
used to perform minimization, and the attributes: 'value', 'counts',
'convergence' and 'message', coming from the \code{\link[stats]{optim}}
method, are added to the result.
}
\description{
The Lientz mode estimator is nothing but the value minimizing the empirical
Lientz function. A 'plot' and a 'print' methods are provided.
}
\details{
The Lientz function is the smallest non-negative quantity \eqn{S(x,\beta)}{S(x,b)},
where \eqn{\beta}{b} = \code{bw}, such that
\deqn{F(x+S(x,\beta)) - F(x-S(x,\beta)) \geq \beta.}{F(x+S(x,b)) - F(x-S(x,b)) >= b.}
Lientz (1970) provided a way to estimate \eqn{S(x,\beta)}{S(x,b)}; this estimate
is what we call the empirical Lientz function.
}
\note{
The user may call \code{mlv.lientz} through
\code{mlv(x, method = "lientz", ...)}.
}
\examples{
# Unimodal distribution
x <- rbeta(1000,23,4)
## True mode
betaMode(23, 4)
## Lientz object
f <- lientz(x, 0.2)
print(f)
plot(f)
## Estimate of the mode
mlv(f) # optim(shorth(x), fn = f)
mlv(f, abc = TRUE) # x[which.min(f(x))]
mlv(x, method = "lientz", bw = 0.2)
# Bimodal distribution
x <- c(rnorm(1000,5,1), rnorm(1500, 22, 3))
f <- lientz(x, 0.1)
plot(f)
}
\references{
\itemize{
\item Lientz B.P. (1969).
On estimating points of local maxima and minima of density functions.
\emph{Nonparametric Techniques in Statistical Inference (ed. M.L. Puri, Cambridge University Press}, p.275-282.
\item Lientz B.P. (1970).
Results on nonparametric modal intervals.
\emph{SIAM J. Appl. Math.}, \bold{19}:356-366.
\item Lientz B.P. (1972).
Properties of modal intervals.
\emph{SIAM J. Appl. Math.}, \bold{23}:1-5.
}
}
\seealso{
\code{\link[modeest]{mlv}} for general mode estimation;
\code{\link[modeest]{shorth}} for the shorth estimate of the mode
}
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