Density Estimation using Adaptive Kernel method
Univariate adaptive kernel density estimation a la Silverman. As used by Portnoy and Koenker (1989).
akj(x, z =, p =, h = -1, alpha = 0.5, kappa = 0.9, iker1 = 0)
x |
points used for centers of kernel assumed to be sorted. |
z |
points at which density is calculated; defaults to an equispaced sequence covering the range of x. |
p |
vector of probabilities associated with |
h |
initial window size (overall); defaults to Silverman's normal reference. |
alpha |
a sensitivity parameter that determines the sensitivity of the local bandwidth to variations in the pilot density; defaults to .5. |
kappa |
constant multiplier for initial (default) window width |
iker1 |
integer kernel indicator: 0 for normal kernel (default)
while 1 for Cauchy kernel ( |
a list structure is with components
dens |
the vector of estimated density values f(z) |
psi |
a vector of ψ=-f'/f function values. |
score |
a vector of score ψ' = (f'/f)^2 - f''/f function values. |
h |
same as the input argument h |
if the score function values are of interest, the Cauchy kernel
may be preferable.
Portnoy, S and R Koenker, (1989) Adaptive L Estimation of Linear Models; Annals of Statistics 17, 362–81.
Silverman, B. (1986) Density Estimation, pp 100–104.
set.seed(1)
x <- c(rnorm(600), 2 + 2*rnorm(400))
xx <- seq(-5, 8, length=200)
z <- akj(x, xx)
plot(xx, z$dens, ylim=range(0,z$dens), type ="l", col=2)
abline(h=0, col="gray", lty=3)
plot(xx, z$psi, type ="l", col=2, main = expression(hat(psi(x))))
plot(xx, z$score, type ="l", col=2,
main = expression("score " * hat(psi) * "'" * (x)))
if(require("nor1mix")) {
m3 <- norMix(mu= c(-4, 0, 3), sig2 = c(1/3^2, 1, 2^2),
w = c(.1,.5,.4))
plot(m3, p.norm = FALSE)
set.seed(11)
x <- rnorMix(1000, m3)
z2 <- akj(x, xx)
lines(xx, z2$dens, col=2)
z3 <- akj(x, xx, kappa = 0.5, alpha = 0.88)
lines(xx, z3$dens, col=3)
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