MM-estimates for Regression

In 1987 Yohai introduced the MM-estimates for regression. Assume the regression model (8) and recall that G denotes the distribution of $\epsilon_i$. Consider two functions $\rho_0, \rho_1: {\mbox{I}\!\mbox{R}}\rightarrow {\mbox{I}\!\mbox{R}}$ such that

$$E_G \left( \rho_0 \left( \epsilon \right) \right) = 0.5, \ \ ... ... = \sup_{x \in {\mbox{I}\!\mbox{R}}} \rho_1 \left( x \right)$$

The MM-estimate ${\bf T}^1$ is defined in three steps as follows:

1.

Compute an initial regression estimate ${\bf T}^0$

2.

Compute the M-scale of the residuals $r_i \left( {\bf T}^0 \right)$, S_n, using the function $\rho_0$.

3.

Define ${\bf T}^1$ as any solution to the equation

$$\sum_{i=1}^n\rho_1' \left( r_i \left( {\bf T}^1 \right) / S_n \right) {\bf x}_i = 0$$

that also satisfies

$$\sum_{i=1}^n\rho_1 \left( r_i \left( {\bf T}^1 \right) / S_n ... ...1}^n\rho_1 \left( r_i \left( {\bf T}^0 \right) / S_n \right).$$

These estimates have high breakdown point and high efficiency at the central model. Under the usual regularity conditions, including that the distribution of the errors is symmetric, these estimates are strongly consistent and asymptotically normal, with variance depending only on the limiting value of the scale estimator S_n.