Table 1. Comparison between OLS estimator and M–Estimators.

Objective ρ(e) and weight w(e) functions of OLS solution and five different M–Estimators. For M–Estimators, error entries are standardized, i.e. divided by the scale estimator, s. Each robust variant comes with a hyperparameter c. Exemplary plots in the last column utilize the optimal hyperparameters detailed in Table 2.

Type	ρ(e)	w(e)	w(e)
OLS	$\frac{1}{2}{e}^2$	1
Cauchy	$\frac{c^2}{2}log\left(1+{\left(\frac{e}{c}\right)}^2\right)$	$\frac{1}{1+{\left(e∕c\right)}^2}$
Fair	$c^2\left(\right.\frac{\left|e\right|}{c}-log\left(\right.1+\frac{\left|e\right|}{c}\left)\right.\left)\right.$	$\frac{1}{1+\left|e\right|∕c}$
Huber  $${\displaystyle \left\{\right.\begin{array}{c}\text{if}\left|e\right|\le c\hfill \\ \text{if}\left|e\right|\ge c\hfill \\ \hfill \end{array}}$$	$${\displaystyle \left\{\right.\begin{array}{c}\frac{e^2}{2}\hfill \\ c\left(\right.\left|e\right|-\frac{c}{2}\left)\right.\hfill \\ \hfill \end{array}}$$	$${\displaystyle \left\{\right.\begin{array}{c}1\hfill \\ \frac{c}{\left|e\right|}\hfill \\ \hfill \end{array}}$$
Tukey $${\displaystyle \left\{\right.\begin{array}{c}\text{if}\left|e\right|\le c\hfill \\ \text{if}\left|e\right|>c\hfill \\ \hfill \end{array}}$$	$${\displaystyle \left\{\right.\begin{array}{c}\frac{c^2}{6}\left(\right.1-\left(\right.1-\left(\right.\frac{e}{c}{\left)\right.}^2{\left)\right.}^3\left)\right.\hfill \\ \frac{c^2}{6}\hfill \\ \hfill \end{array}}$$	$${\displaystyle \left\{\right.\begin{array}{c}\left(\right.1-{\left(\frac{e}{c}\right)}^2{\left)\right.}^2\hfill \\ 0\hfill \\ \hfill \end{array}}$$
Welsch	$\frac{c^2}{2}\left(1-exp\left(-{\left(\frac{e}{c}\right)}^2\right)\right)$	$exp\left(-{\left(\frac{e}{c}\right)}^2\right)$