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\name{logConDens}
\alias{logConDens}
\title{Compute log-concave density estimator and related quantities}
\description{Compute the log-concave and smoothed log-concave density estimator.}
\usage{logConDens(x, xgrid = NULL, smoothed = TRUE, print = FALSE,
gam = NULL, xs = NULL)}
\arguments{
\item{x}{Vector of independent and identically distributed numbers, not necessarily unique.}
\item{xgrid}{Governs the generation of weights for observations. See \code{\link{preProcess}} for details.}
\item{smoothed}{If \code{TRUE}, the smoothed version of the log-concave density estimator is also computed.}
\item{print}{\code{print = TRUE} outputs the log-likelihood in every loop, \code{print = FALSE} does not. Make sure to tell \code{R} to output (press CTRL+W).}
\item{gam}{Only necessary if \code{smoothed = TRUE}. The standard deviation of the normal kernel. If equal to \code{NULL}, \code{gam} is chosen
such that the variances of the original sample \eqn{x_1, \ldots, x_n} and \eqn{\widehat f_n^*}{\hat f_n^*} coincide.}
\item{xs}{Only necessary if \code{smoothed = TRUE}. Either provide a vector of support points where the smoothed estimator should be computed at, or
leave as \code{NULL}. Then, a sufficiently width equidistant grid of points will be used.}
}
\details{See \code{\link{activeSetLogCon}} for details on the computations.}
\value{
\code{\link{logConDens}} returns an object of class \code{"dlc"}, a list containing the
following components:
\code{xn}, \code{x}, \code{w}, \code{phi}, \code{IsKnot}, \code{L}, \code{Fhat}, \code{H},
\code{n}, \code{m}, \code{knots}, \code{mode}, and \code{sig} as generated
by \code{\link{activeSetLogCon}}. If \code{smoothed = TRUE}, then the returned object additionally contains
\code{f.smoothed}, \code{F.smoothed}, \code{gam}, and \code{xs} as generated by \code{\link{evaluateLogConDens}}. Finally, the
entry \code{smoothed} of type \code{"logical"} returnes the value of \code{smoothed}.
The methods \code{\link{summary.dlc}} and \code{\link{plot.dlc}} are used to obtain a summary and generate plots of the estimated
density.
}
\references{
Duembgen, L, Huesler, A. and Rufibach, K. (2010).
Active set and EM algorithms for log-concave densities based on complete and censored data.
Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}.
Duembgen, L. and Rufibach, K. (2009).
Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency.
\emph{Bernoulli}, \bold{15(1)}, 40--68.
Duembgen, L. and Rufibach, K. (2011).
logcondens: Computations Related to Univariate Log-Concave Density Estimation.
\emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06}
}
\author{
Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch}
Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}
}
\examples{
## ===================================================
## Illustrate on simulated data
## ===================================================
## Set parameters
n <- 50
x <- rnorm(n)
res <- logConDens(x, smoothed = TRUE, print = FALSE, gam = NULL,
xs = NULL)
summary(res)
plot(res, which = "density", legend.pos = "topright")
plot(res, which = "log-density")
plot(res, which = "CDF")
## Compute slopes and intercepts of the linear functions that
## compose phi
slopes <- diff(res$phi) / diff(res$x)
intercepts <- -slopes * res$x[-n] + res$phi[-n]
## ===================================================
## Illustrate method on reliability data
## Reproduce Fig. 2 in Duembgen & Rufibach (2009)
## ===================================================
## Set parameters
data(reliability)
x <- reliability
n <- length(x)
res <- logConDens(x, smooth = TRUE, print = TRUE)
phi <- res$phi
f <- exp(phi)
## smoothed log-concave PDF
f.smoothed <- res$f.smoothed
xs <- res$xs
## compute kernel density
sig <- sd(x)
h <- sig / sqrt(n)
f.kernel <- rep(NA, length(xs))
for (i in 1:length(xs)){
xi <- xs[i]
f.kernel[i] <- mean(dnorm(xi, mean = x, sd = h))
}
## compute normal density
mu <- mean(x)
f.normal <- dnorm(xs, mean = mu, sd = sig)
## ===================================================
## Plot resulting densities, i.e. reproduce Fig. 2
## in Duembgen and Rufibach (2009)
## ===================================================
plot(0, 0, type = 'n', xlim = range(xs), ylim = c(0, 6.5 * 10^-3))
rug(res$x)
lines(res$x, f, col = 2)
lines(xs, f.normal, col = 3)
lines(xs, f.kernel, col = 4)
lines(xs, f.smoothed, lwd = 3, col = 5)
legend("topleft", c("log-concave", "normal", "kernel",
"log-concave smoothed"), lty = 1, col = 2:5, bty = "n")
## ===================================================
## Plot log-densities
## ===================================================
plot(0, 0, type = 'n', xlim = range(xs), ylim = c(-20, -5))
legend("bottomright", c("log-concave", "normal", "kernel",
"log-concave smoothed"), lty = 1, col = 2:5, bty = "n")
rug(res$x)
lines(res$x, phi, col = 2)
lines(xs, log(f.normal), col = 3)
lines(xs, log(f.kernel), col = 4)
lines(xs, log(f.smoothed), lwd = 3, col = 5)
## ===================================================
## Confidence intervals at a fixed point for the density
## see help file for logConCI()
## ===================================================
}
\keyword{htest}
\keyword{nonparametric}
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