1\name{akj}
2\alias{akj}
3\title{Density Estimation using Adaptive Kernel method}
4\description{
5  Univariate \emph{adaptive} kernel density estimation a la Silverman.
6  As used by Portnoy and Koenker (1989).
7}
8\usage{
9akj(x, z =, p =, h = -1, alpha = 0.5, kappa = 0.9, iker1 = 0)
10}
11\arguments{
12  \item{x}{points used for centers of kernel assumed to be sorted.}
13  \item{z}{points at which density is calculated; defaults to an
14    equispaced sequence covering the range of x.}
15  \item{p}{vector of probabilities associated with \code{x}s; defaults
16    to 1/n for each x.}
17  \item{h}{initial window size (overall); defaults to Silverman's normal
18    reference.}
19  \item{alpha}{a sensitivity parameter that determines the sensitivity of
20    the local bandwidth to variations in the pilot density; defaults to .5.}
21  \item{kappa}{constant multiplier for initial (default) window width}
22  \item{iker1}{integer kernel indicator: 0 for normal kernel (default)
23    while 1 for Cauchy kernel (\code{\link{dcauchy}}).}
24}
25\value{
26  a \code{\link{list}} structure is with components
27  \item{dens}{the vector of estimated density values \eqn{f(z)}}
28  \item{psi}{a vector of \eqn{\psi=-f'/f} function values.}
29  \item{score}{a vector of score \eqn{\psi' = (f'/f)^2 - f''/f} function
30    values.}
31  \item{h}{same as the input argument h}
32}
33\note{
34  if the \code{score} function values are of interest, the Cauchy kernel
35  may be preferable.
36}
37\references{
38  Portnoy, S and R Koenker, (1989)
39  Adaptive L Estimation of Linear Models;
40  \emph{Annals of Statistics} \bold{17}, 362--81.
41
42  Silverman, B. (1986)
43  \emph{Density Estimation}, pp 100--104.
44}
45\examples{
46 set.seed(1)
47 x <- c(rnorm(600), 2 + 2*rnorm(400))
48 xx <- seq(-5, 8, length=200)
49 z <- akj(x, xx)
50 plot(xx, z$dens, ylim=range(0,z$dens), type ="l", col=2)
51 abline(h=0, col="gray", lty=3)
52 plot(xx, z$psi, type ="l", col=2, main = expression(hat(psi(x))))
53 plot(xx, z$score, type ="l", col=2,
54      main = expression("score " * hat(psi) * "'" * (x)))
55
56 if(require("nor1mix")) {
57  m3 <- norMix(mu= c(-4, 0, 3), sig2 = c(1/3^2, 1, 2^2),
58               w = c(.1,.5,.4))
59  plot(m3, p.norm = FALSE)
60  set.seed(11)
61  x <- rnorMix(1000, m3)
62  z2 <- akj(x, xx)
63  lines(xx, z2$dens, col=2)
64  z3 <- akj(x, xx, kappa = 0.5, alpha = 0.88)
65  lines(xx, z3$dens, col=3)
66 }
67}
68\keyword{smooth}
69