Holland, P. W. and Welsch, R. E. (1977), "Robust regression using iteratively reweighted least-squares", Communications in Statistics - Theory and Methods 6(9), 813-827, doi:10.1080/03610927708827533. The bibliographic record is verified against Crossref.
Details
CITATION LIMIT, stated rather than papered over. The article is closed
access: Taylor and Francis returns 403 to every fetch, OpenAlex and Semantic
Scholar report no open-access location, and no preprint or technical-report
version was found. No page or equation number is therefore attributed to it.
The weight function below is the one this module was specified with,
w(r) = 1/(1 + (r/c)^2), and the companions follow from it by the
standard M-estimation relations rather than by quotation:
psi(r) = r w(r) = r/(1 + (r/c)^2) and
rho(r) = (c^2/2) log(1 + (r/c)^2), with d rho / dr = psi.
The rho is the log-density of a Cauchy up to constants, which is where the name comes from and why the function behaves as it does. Unlike Huber, the weight starts falling at r = 0 and never reaches zero: every observation keeps some influence, however far out it lies, so the estimator redescends but does not reject. psi attains its maximum at r = c and decreases thereafter, so a residual twice as large as c carries LESS weight in absolute terms than one at c – the property Huber lacks and the bisquare takes to the extreme of exact rejection.
The default tuning constant c = 2.3849 is the value conventionally quoted for about 95 percent asymptotic efficiency at the Gaussian; the function does not assert it, and the accompanying anchor derives the efficiency \((E psi')^2 / E psi^2\) by quadrature instead of trusting the number.
References
Holland, P. W. and Welsch, R. E. (1977), Communications in Statistics - Theory and Methods 6(9):813-827, doi:10.1080/03610927708827533.
Examples
Cauchw(c(0, 1, 2.3849, 10))$weights # 1, ..., exactly 0.5 at r = c, ...
#> [1] 1.00000000 0.85047284 0.50000000 0.05381653