J1: J1
In kader: Kernel Adaptive Density Estimation and Regression

Description Usage Arguments Details Value Note See Also Examples

Eq. (15.16) in Eichner (2017) as a result of Cardano's formula.

1	J1(u, cc = sqrt(5/3))

`u`	Numeric vector.
`cc`	Numeric constant, defaults to √(5/3).

Using, for brevity's sake, J_{1a}(u, c) := -q_c(u) and J_{1b}(u, c) := J_{1a}(u, c)^2 + p_c^3, the definition of J_1 reads:

J_1(u, c) := [J_{1a}(u, c) + √(J_{1b}(u, c))]^{1/3} + [J_{1a}(u, c) - √(J_{1b}(u, c))]^{1/3}.

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).

Vector of same length and mode as u.

Eq. (15.16) in Eichner (2017), and hence J_1(u, c), requires c to be in [√(5/3), 3). If cc does not satisfy this requirement a warning (only) is issued.

J_admissible.

u <- seq(0, 1, by = 0.01)
c0 <- expression(sqrt(5/3))
c1 <- expression(sqrt(3) - 0.01)
cgrid <- c(1.35, seq(1.4, 1.7, by = 0.1))
cvals <- c(eval(c0), cgrid, eval(c1))

Y <- sapply(cvals, function(cc, u) J1(u, cc = cc), u = u)
cols <- rainbow(ncol(Y), end = 9/12)
matplot(u, Y, type = "l", lty = "solid", col = cols,
  ylab = expression(J[1](u, c)))
abline(h = 0)
legend("topleft", title = "c", legend = c(c0, cgrid, c1),
  lty = 1, col = cols, cex = 0.8)