Gaussian-shaped weight that decays smoothly and never reaches zero

Unlike the biweight there is no hard cutoff – a residual ten times the tuning constant still counts, just by e^-100. The objective is smooth everywhere, so gradient methods behave where the biweight kink at c can stall them.

Usage

Welshw(y, c = 2.9846)

Arguments

y

Scaled residuals.

c

Tuning constant; 2.9846 gives 95 percent Gaussian efficiency.

Value

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

Details

Formula: w(r) = exp(-(r/c)^2), rho(r) = (c^2/2)[1 - exp(-(r/c)^2)], psi = r w(r).

References

Dennis, J. E. & Welsch, R. E. (1978). Commun Statist B 7:345-359.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Welshw(V)
#> $estimate
#> [1] 26.07811
#> 
#> $w
#> [1] 0.8938112141 0.6382387915 0.3640928853 0.1659330082 0.0604150277
#> [6] 0.0175731158 0.0040836091 0.0007581081
#> 
#> $rho
#> [1] 0.4729562 1.6112550 2.8322785 3.7148665 4.1848350 4.3756494 4.4357305
#> [8] 4.4505420
#> 
#> $psi
#> [1] 0.893811214 1.276477583 1.092278656 0.663732033 0.302075139 0.105438695
#> [7] 0.028585263 0.006064865
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Welsch robust weight"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"