# J2: J2 In kader: Kernel Adaptive Density Estimation and Regression

## Description

Eq. (20) in Eichner (2017) (based on "Bronstein's formula for k = 3")

## Usage

 `1` ```J2(u, cc = sqrt(5)) ```

## Arguments

 `u` Numeric vector. `cc` Numeric constant, defaults to √ 5.

## Details

J_2(u, c) = 2√(-p_c) * sin(1/3 * arcsin(q_c(u) / (-p_c)^{3/2}))

For implementation details of q_c(u) and p_c see `qc` and `pc`, respectively.

For further mathematical details see Eichner (2017) and/or Eichner & Stute (2013).

## Value

Vector of same length and mode as `u`.

## Note

Eq. (20) in Eichner (2017), and hence J_2(u, c), requires c to be in (√ 3, √ 5]. If `cc` does not satisfy this requirement (only) a warning is issued.

The default `cc = sqrt(5)` yields the optimal rank transformation.

## See Also

`J_admissible`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```u <- seq(0, 1, by = 0.01) c0 <- expression(sqrt(3) + 0.01) c1 <- expression(sqrt(5)) cgrid <- seq(1.85, 2.15, by = 0.1) cvals <- c(eval(c0), cgrid, eval(c1)) Y <- sapply(cvals, function(cc, u) J2(u, cc = cc), u = u) cols <- rainbow(ncol(Y), end = 9/12) matplot(u, Y, type = "l", lty = "solid", col = cols, ylab = expression(J(u, c))) abline(h = 0) legend("topleft", title = "c", legend = c(c0, cgrid, c1), lty = 1, col = cols, cex = 0.8) ```

kader documentation built on May 1, 2019, 10:13 p.m.