3.13 Robust Regression: 3.13.1 M-Estimates, 3.13.2 Estimates Based on Robust Location and Scale Measures, 3.13.3 Measuring Robustness, 3.13.4 Other Robust Estimates

Let $\chi (z) = \operatorname {sign}(\left|z\right|-1 / c)$ for some constant $c$. Show that the solution of (3.77) is the MAD estimate

Show that if we put $w_{i j} = 1$ in (3.99), we get Jaeckel's estimator defined by (3.97) with Wilcoxon weights.

Show that if $s$ is the solution of (3.101) with $\rho$ given by (3.102), then the resulting $s$ is approximately the lower quartile of the $\binom {n}{2}$ differences $\left|e_{i}(\mathbf{b})-e_{j}(\mathbf{b})\right|$.