M-estimators

\[\rho(x)=\frac{1}{2}x^2\]

\[\rho(x)=|x|\]

\[\rho(x)=\frac{1}{2}x^2~ 1\{|x| \leq c\}\]

\[\rho(x)= \left\{ \begin{array}{ll}\frac{1}{2}x^2, & |x| \leq c\\ c(|x|-\frac{1}{2}c), & |x|>c \end{array} \right.\]

\[\psi(x)= sign(x)\cdot\left\{ \begin{array}{ll} |x|, & |x|<c_1 \\ c_1, & c_1 \leq |x| \leq c_2 \\ \frac{c_1(c_3-|x|)}{c_3-c_2}, & c_2 < |x| < c_3 \\ 0, & |x| \geq c_3 \end{array} \right.\]

\[w(x)=\left(1-\left(\frac{x}{c}\right)^2\right)_+^2\]