The biweight loss and its derivative, evaluated pointwise

rho is the loss an M-estimator minimises, psi its derivative, w the IRLS weight psi(r)/r. rho is flat past c, so a residual beyond the tuning constant costs a constant and the fit stops chasing it.

Usage

Tukrho(r, c = 4.685)

Arguments

r: Scaled residuals.
c: Tuning constant.

Value

List with estimate, rho, psi, w, n.

Details

Formula: rho(r) = (c^2/6)(1 - [1 - (r/c)^2]^3) for |r| <= c, else c^2/6; psi(r) = r[1 - (r/c)^2]^2 inside and 0 outside.

References

Beaton, A. E. & Tukey, J. W. (1974). Technometrics 16:147-185.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Tukrho(V)
#> $estimate
#> [1] 23.26043
#> 
#> $rho
#> [1] 0.4775661 1.6576631 2.9070282 3.5853605 3.6582042 3.6582042 3.6582042
#> [8] 3.6582042
#> 
#> $psi
#> [1] 0.9109563 1.3374668 1.0441681 0.2938613 0.0000000 0.0000000 0.0000000
#> [8] 0.0000000
#> 
#> $w
#> [1] 0.91095630 0.66873341 0.34805604 0.07346533 0.00000000 0.00000000 0.00000000
#> [8] 0.00000000
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Tukey biweight rho, psi and weight"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"