# akj

0th

Percentile

##### Density Estimation using Adaptive Kernel method

Univariate adaptive kernel density estimation a la Silverman. As used by Portnoy and Koenker (1989).

Keywords
smooth
##### Usage
akj(x, z =, p =, h = -1, alpha = 0.5, kappa = 0.9, iker1 = 0)
##### Arguments
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 xs; defaults to 1/n for each x.

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 (dcauchy).

##### Value

a list structure is with components

dens

the vector of estimated density values $f(z)$

psi

a vector of $\psi=-f'/f$ function values.

score

a vector of score $\psi' = (f'/f)^2 - f''/f$ function values.

h

same as the input argument h

##### Note

if the score function values are of interest, the Cauchy kernel may be preferable.

##### References

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.

• akj
##### Examples
# NOT RUN {
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)
}
# }

Documentation reproduced from package quantreg, version 5.54, License: GPL (>= 2)

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