Yohai and Zamar (1997) showed that the function given above
is optimal in the following highly desirable sense: the final M estimate
has a breakdown point of one-half and minimizes the maximum bias under
contamination distributions (locally for small fraction of
contamination), subject to achieving a desidered nominal asymptotic
efficiency when the data are Gaussian.
\label{opt}
\rho(x) = \begin{cases}
1.3846 \left( \frac{x}{c} \right)^2 \qquad |x| \leq \frac{2}{3} c \\
0.5514-2.6917\left( \frac{x}{c} \right)^2+10.7668\left( \frac{x}{c} \right)^4-11.6640\left( \frac{x}{c} \right)^6+4.0375\left( \frac{x}{c} \right)^8
\qquad \frac{2}{3} c < |x| \leq c
\\
1 \qquad |x| >c
\end{cases}