Module scale 

The ScaleEstimator trait and standard robust scale estimates.

Structs

HuberProposal2

Huber’s Proposal 2 scale: solve (1/n) Σ ψ((rᵢ−μ)/s)² = β (with β = E_Φ[ψ²]) jointly with the location, by fixed-point iteration.

Mad

MAD: s = 1.4826 · median(|rᵢ − median(r)|). The constant 1.4826 = 1/Φ⁻¹(¾) makes it consistent for σ at the Gaussian model.

Qn

Qn = c · dₙ · {|rᵢ − rⱼ| : i < j}₍ₖ₎, the k-th order statistic of the pairwise absolute differences with k = C(⌊n/2⌋+1, 2) (≈ the ¼-quantile). The constant c = 2.2219 = 1/(√2·Φ⁻¹(⅝)) gives consistency for σ at the Gaussian; dₙ is a finite-sample correction (Croux & Rousseeuw 1992).

SScale

The M-estimate of scale (“S-scale”): the s > 0 solving (1/n) Σ ρ(rᵢ/s) = δ.

Sn

Sn = c · dₙ · med_i { med_j |rᵢ − rⱼ| }: a low-median over points of the per-point high-median of the pairwise absolute differences. The constant c = 1.1926 gives consistency for σ at the Gaussian; dₙ is a finite-sample correction (Croux & Rousseeuw 1992).

Traits

ScaleEstimator

Estimates the scale s used to standardize residuals as r / s. Without a scale estimate the tuning constants are meaningless.