Siegel, A. F. (1982), "Robust regression using repeated medians", Biometrika 69(1), 242-244, doi:10.1093/biomet/69.1.242.
Value
List with estimate (the slope), slope,
intercept, fitted, residuals, n,
n_used, breakdown_point, method.
Details
CITATION LIMIT, stated rather than papered over. The Biometrika article is
closed access (Semantic Scholar and OpenAlex both report no open-access
location) and the Princeton technical-report version, DTIC ADA092660,
returns 403 to every fetch tried. The estimator is therefore taken from a
source that states it in full and cites Siegel for it: Borowski, M. and
Fried, R. (2011), "Robust repeated median regression in moving windows with
data-adaptive width selection", SFB 823 Discussion Paper 28/2011, TU
Dortmund University, doi:10.17877/de290r-13059, Section 2, equation (3),
page 3, read off a rendered page image. That equation is
\(eta = med_i med_{i' != i} (y_i - y_i')/(x_i - x_i')\) and
level = med_i (y_i - beta x_i), and the same page states the property
that makes the estimator worth having: a finite-sample replacement breakdown
point of floor(n/2)/n, about 50 percent, which Davies and Gather
(2005) show is the maximum possible for a regression-equivariant estimator.
The doubled median is the whole idea. Theil-Sen takes ONE median over all n(n-1)/2 pairwise slopes and breaks down at about 29 percent; taking a median within each point first, and then across points, means a minority of arbitrarily corrupted observations cannot move either level of the calculation. Half the sample can be replaced by nonsense and the line does not move to infinity.
Points sharing an x value with no other point contribute no inner median and are skipped; if no pair of distinct x values exists at all, no slope is defined and that is an error rather than a number.