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Garc´ıa Ben et al. [8] extended τ -estimates to the MLM by defining 

subject to 

where the τ -scale is defined by 

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where v = ( v$_{1}$, . . . , v$_{n}$ ), ρ is a ρ -function and s is an M-estimate of scale. 

The robust estimates are based on bisquare ρ -functions. The M-estimate of scale used in the S-estimate is defined by ρ$_{0}$ ( u ) = ρ$_{B}$ ( u/c$_{0}$ ), and b = 0 . 5 so that the S-estimate has breakdown point 0 . 5 (see Table 1). The τ -estimate uses the same ρ$_{0}$ and b as the S-estimate to compute the M-scale and ρ$_{2}$ ( u ) = ρ$_{B}$ ( u/c$_{2}$ ), where c$_{2}$ is chosen together with the constant κ , from equation (8.1), so that the τ -estimate has an ARE equal to 0 . 90 when the errors are Gaussian (see Table 2 in [8] in which κ = 6 κ$_{2}$/c 2 $_{2}$). The initial estimate needed to compute the τ -estimate is computed using 2000 subsamples. The MM-estimate uses the same ρ$_{0}$ as the S-estimate to compute the M-estimate of scale and ρ$_{1}$ ( u ) = ρ$_{B}$ ( u/c$_{1}$ ), where c$_{1}$ is chosen so that the MM-estimate has an ARE equal to 0 . 90 when the errors are Gaussian (see Table 2). We use the S-estimates as ( ˜ B$_{n}$ , ˜ Σ$_{n}$ ) and the value of δ in step 5 of the computing algorithm is taken equal to 10 − $^{4}$. 

