Support-preservation (basin) radius for a fixed-effects start.
ransac_basin_radius(object, cc = NULL, rho = NULL)the basin radius \(r^\star(c)\) in the units of
\(\beta\) (a scalar), with attribute "max_xnorm".
a fitted lmerMod or rlmerMod supplying
the error scale \(\sigma\) and the fixed-effects design matrix.
the rejection point \(c\) of the redescender. If
NULL and rho is supplied, \(c\) is the rejection
point of rho found numerically; if both are NULL, the
bisquare default 4.685.
an optional redescending psi_func_rcpp (e.g.
bisquarePsi, lqqPsi) whose rejection
point supplies \(c\) when cc is NULL.
Computes the radius \(r^\star(c) = c\,\sigma / (2 \max_j
\|x_j\|)\) of the ball around the initial fixed-effects estimate
within which every observation keeps its redescender-support
membership, so the population Hessian stays positive definite (Koller
and Stahel; the RANSAC-RSE basin theorem). Here \(c\) is the
rejection point of the redescending \(\psi\) --- the smallest
\(x > 0\) with \(\psi(x) = 0\). For the bisquare this is the tuning
cutoff (default 4.685); the geometry generalises to any
finite-rejection-point redescender (e.g. lqqPsi) by
finding its rejection point numerically. A redescending \(\psi\) is
safe to engage from a start only if the eventual estimate stays within
this radius of the (high-breakdown) start --- otherwise the iteration
may leave the basin and converge to a phony solution.