$\tau$-estimates for Regression

In 1988, Yohai and Zamar introduced the class of $\tau$-estimates. Assume the linear model (8) and consider two functions $\rho_1, \rho_2: {\mbox{I}\!\mbox{R}}\rightarrow {\mbox{I}\!\mbox{R}}$. For each $\beta \in {\mbox{I}\!\mbox{R}}^p$ define the $\tau$ estimate of the scale of the residuals $\left( r_1 \left( \beta \right), \ldots, r_n \left( \beta \right) \right)$ by

$$\tau^2_n \left( \beta \right) = s_n^2 \left( \beta \right)... ...rac{r_i\left(\beta\right)}{s_n \left( \beta \right)} \right)$$

where $s_n \left( \beta \right)$ is the M-scale of the residuals calculated using the function $\rho_1$. The $\tau$-estimates for regression are then defined as

$$\hat{\theta} = \arg \min_{\beta} \tau_n \left( \beta \right).$$

In particular, $\tau_n (\hat{\theta})$ is an efficient and robust estimate of $\sigma$. As before, under certain regularity conditions (which include symmetry of the distribution of the residuals) we get strong consistency of these estimates and asymptotic normality.