Robust estimators for FA

2. Robust estimators for FA

2.1. Two-stage estimator To estimate µ and Σ, we use an S -estimator with a high breakdown point. Very generally, an S -estimator of multivariate location and scale is defined as the solution for µ and Σ that minimizes the determinant of Σ, i.e. |Σ|, subject to 1 n n X i=1 ρ q x i − µ T Σ −1 x i − µ = 1 n n X i=1 ρd i = b , 4 where ρ is a symmetric, continuously differentiable function, with ρ0 = 0 and ρ is strictly increasing on [0, c] and constant on [c, ∞] and c is such that E Φ [ρ]ρc = ǫ ∗ where Φ is the standard normal and ǫ ∗ is the chosen breakdown point see Rousseeuw and Yohai, 1984 . b is a parameter chosen to determine the breakdown point by means of b = ǫ ∗ max d ρd. For ρ, we use the biweight ρ-function of Tukey. Hence, the biweight S -estimator BS is defined by the minimum of |Σ| subject to 4, with ρd; c =        d 2 2 − d 4 2c 2 + d 6 6c 4 , ≤ d ≤ c c 2 6. d c 5 Estimating equations for the BS -estimator are µ = 1 P n i=1 u d i ; c n X i=1 u d i ; c x i , 6 Σ = p P n i=1 u d i ; c d 2 i n X i=1 u d i ; c x i − µ x i − µ T , with weights u d; c = 1 d ∂ ∂d ρd; c =        1 − dc 2 2 , ≤ d ≤ c 0. d c The BS -estimator of Σ can then be used to replace S or ˆσ in the objective functions 2 or 3; this defines the two-stage estimator MLE or GLS. 2.2. Direct estimator The FA model implies that the data have been generated by a multivariate normal distribution with constrained covariance matrix according to 1. For this type of problems, one can use the results of Copt and Victoria-Feser 2006 who have developed such an estimator for mixed linear models. In Appendix A, we derive estimating equations for the FA parameters directly from the ob- jective function of the S -estimator, using the Lagrangian of the minimization of |Σ| subject to 4 given by L = log |Σ| + λ        1 n n X i=1 ρd i ; c − b        , 7 3 Let P = h ψ 2 1 , ..., ψ 2 p i T and G = h γ 1 , ..., γ q i T , an iterative system for the γ j and ψ j is given by P t+1 = p n        1 n n X i=1 u d i ; c t d 2t i        −1 C −1t D t , G t+1 = p n        1 n n X i=1 u d i ; c t d 2t i        −1 M −1t N t , with the matrices C, D, M and N given in respectively 26, 27, 29 and 30. Note that µ is calculated at each step with equation 6 and Σ from G and P . For the starting point of this iterative procedure, one can perform a confirmatory FA based on a robust “plug-in” covariance matrix that is fast to compute. Our choice goes to the orthogo- nalized Gnanadesikan-Kettering OGK proposed by Maronna and Zamar 2002. Alternatively, for an even faster solution, one can perform an exploratory FA on e.g. the OGK and set the resulting loadings γ i j that are constrained e.g. to zero to the constrained value e.g. zero. We sucessfully implemented this last alternative and used it for the simulations. The resulting algorithm is as follows 1. Compute µ start = µ OGK and Σ OGK . 2. Compute Ψ start and Γ start by means of an exploratory factor analysis based on Σ OGK . Re- place the fixed γ i j in Γ start by their value e.g. zero. 3. Compute Σ start = Γ start Γ T start + Ψ start . 4. Compute the Mahalanobis distance d 1 i = q x i − µ start T Σ −1 start x i − µ start . 5. Compute the different weights ud 1 i ; c. 6. Compute the mean vector µ 1 = P n i=1 u d 1 i ; cx i P n i=1 u d 1 i ; c .

7. Compute the vector of loadings G and uniqueness P