akj {quantreg} | R Documentation

## Density Estimation using Adaptive Kernel method

### Description

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

### 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 `x`s; 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 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

### 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.

### Examples

     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)
     }

[Package quantreg version 5.36 Index]