ˆ Θ = { n ∑ i =1 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ ) ˆ U$_{i}$ ˆ U T i } − 1 n ∑ i =1 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ ) ˆ U$_{i}$ ˆ V T i , (11) ˆ Σ = 1 n n ∑ i =1 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ ) R$_{i}$R T i , (12) w ( ˆ U$_{i}$ ) =    1 , D 2 i ≤ χ 2 p, 1 − $_{α}$, 0 , otherwise, (13)  where χ 2 p, 1 − α is the 1 − α quantile of the χ 2 p distribution. Another possibility is to use rank-based trimming, The latter will always eliminate the αn observations with largest Mahalanobis distances, even if they are not actual outliers; so we recommend not using an unnecessarily large α w ( ˆ U$_{i}$ ) =     1 , rank( D 2 $_{i}$) /n ≤ 1 − α, 0 , otherwise. (14) where R$_{i}$ = ˆ V$_{i}$ − ˆ Θ T ˆ U$_{i}$ and e$_{i}$ = w ( ˆ U$_{i}$ ) R T i ˆ Σ − $^{1}$R$_{i}$ . These equations can be solved iteratively by a reweighting algorithm. As for the weights w ( ˆ U$_{i}$ ) , they are essentially a by-product of the estimation of µ$_{X}$ , { φ$_{k}$ } and { U$_{i}$ } . Since E ( U$_{i}$ ) = 0 and var( U$_{i}$ ) = diag( λ$_{1}$, . . . , λ$_{p}$ ) , the ˆ U$_{i}$ s are approx- imately uncorrelated with mean zero. The squared Mahalanobis distance of ˆ U$_{i}$ is then D 2 i = ∑ p k =1 ˆ U 2 $_{ik}$/ ˆ λ$_{k}$ , and large D 2 $_{i}$s will correspond to X -outliers. The D 2 $_{i}$s will follow an approximate χ 2 p distribution if the data is Gaussian. This suggests a number of weighting schemes. One possibility is to use “metric” trim- ming, It is shown in the Appendix that ˆ Θ and ˆ Σ satisfy the fixed-point equations 7