tukeyPsi1: Tukey's Bi-square Score (Psi) and "Chi" (Rho) Functions and Derivatives

Description

These are deprecated, replaced by Mchi(*, psi="tukey"), Mpsi(*, psi="tukey")

tukeyPsi1() computes Tukey's bi-square score (psi) function, its first derivative or it's integral/“principal function”. This is scaled such that \(\psi'(0) = 1\), i.e., \(\psi(x) \approx x\) around 0.

tukeyChi() computes Tukey's bi-square loss function, chi(x) and its first two derivatives. Note that in the general context of \(M\)-estimators, these loss functions are called \(\rho (rho)\)-functions.

Usage

tukeyPsi1(x, cc, deriv = 0)
tukeyChi (x, cc, deriv = 0)

Value

a numeric vector of the same length as x.

Arguments

x: numeric vector.
cc: tuning constant
deriv: integer in \(\{-1,0,1,2\}\) specifying the order of the derivative; the default, deriv = 0 computes the psi-, or chi- ("rho"-)function.

Author

Matias Salibian-Barrera, Martin Maechler and Andreas Ruckstuhl

Examples

Run this code

oop <- options(warn = -1) # against the "deprecated" warnings
op <- par(mfrow = c(3,1), oma = c(0,0, 2, 0),
          mgp = c(1.5, 0.6, 0), mar= .1+c(3,4,3,2))
x <- seq(-2.5, 2.5, length = 201)
cc <- 1.55 # as set by default in lmrob.control()
plot. <- function(...) { plot(...); abline(h=0,v=0, col="gray", lty=3)}
plot.(x, tukeyChi(x, cc), type = "l", col = 2)
plot.(x, tukeyChi(x, cc, deriv = 1), type = "l", col = 2)
plot.(x, tukeyChi(x, cc, deriv = 2), type = "l", col = 2)

mtext(sprintf("tukeyChi(x, c = %g, deriv),  deriv = 0,1,2", cc),
      outer = TRUE, font = par("font.main"), cex = par("cex.main"))
par(op)

op <- par(mfrow = c(3,1), oma = c(0,0, 2, 0),
          mgp = c(1.5, 0.6, 0), mar= .1+c(3,4,1,1))
x <- seq(-5, 5, length = 201)
cc <- 4.69 # as set by default in lmrob.control()
plot. <- function(...) { plot(..., asp = 1); abline(h=0,v=0, col="gray", lty=3)}
plot.(x, tukeyPsi1(x, cc), type = "l", col = 2)
abline(0:1, lty = 3, col = "light blue")
plot.(x, tukeyPsi1(x, cc, deriv = -1), type = "l", col = 2)
plot.(x, tukeyPsi1(x, cc, deriv =  1), type = "l", col = 2); abline(h=1,lty=3)

mtext(sprintf("tukeyPsi1(x, c = %g, deriv),  deriv = 0, -1, 1", cc),
      outer = TRUE, font = par("font.main"), cex = par("cex.main"))
par(op)
options(oop)