makeBisquarePsi: Bisquare (Tukey biweight) psi function

In robustlmm: Robust Linear Mixed Effects Models

Bisquare (Tukey biweight) psi function

Description

Construct a redescending Tukey bisquare psi function.

Usage

makeBisquarePsi(c = 4.685)

Arguments

c	tuning cutoff. The default c = 4.685 gives about 95% asymptotic efficiency for the location problem.

Details

Returns a psi_func_rcpp object whose psi, rho, weight and derivative slots implement Tukey's bisquare (biweight) function

\psi(x) = x (1 - (x/c)^2)^2 \quad \text{for } |x| \le c,\quad 0 \text{ otherwise}

suitable for use as rho.e (or rho.b, rho.sigma.e, rho.sigma.b) in rlmer.

Since robustlmm 3.5.0 the psi, rho, weight and derivative evaluations delegate to robustbase's compiled bisquare family (Mpsi, Mwgt, Mchi), so they match lmrob exactly; the returned values are identical (to numerical tolerance) to the previous hand-coded implementation. makeBisquarePsi is the scalar-cutoff special case of the general makeRobustbasePsi.

The Fisher consistency expectations (Erho, Epsi2, EDpsi) are computed by numerical integration against the standard normal.

The bisquare redescends comparatively fast; for a redescending fit the "lqq" psi (lqqPsi), the recommended redescender of robustbase's lmrob.control(setting = "KS2014"), is generally preferable (Koller and Stahel 2011).

Value

psi_func_rcpp object usable in rho.e, rho.b.

References

Koller, M. and Stahel, W. A. (2011) Sharpening Wald-type inference in robust regression for small samples. Computational Statistics & Data Analysis 55(8), 2504–2515.

See Also

makeRobustbasePsi, lqqPsi.

Examples

  pf <- bisquarePsi
  pf@psi(c(-6, -3, 0, 3, 6))

robustlmm documentation built on July 30, 2026, 5:11 p.m.