Λ = diag(1 , 1 / 2 , . . . , 1 /p ) and Σ = diag(1 , 1 / 2 , . . . , 1 /q ) . Two regression parameters Θ$_{0}$ were considered: for the first set of simulations (to study es timation error) we took Θ$_{0}$ with θ$_{0}$$_{,}$$_{11}$ = 3 and θ$_{0}$$_{,ij}$ = 0 for ( i, j ) = (1 , 1) ; for the second set of simulations (to study the goodness of the asymptotic approximation of Wald’s test) we took Θ$_{0}$ = O . The curves { X$_{i}$ ( s ) } and { Y$_{i}$ ( t ) } were generated following (3) and (4), with µ$_{X}$ ( s ) and µ$_{Y}$ ( t ) equal to zero, φ$_{k}$ ( s ) = √$_{2 sin(}$$_{kπs}$$_{)}$ and ψ$_{l}$ ( t ) = √$_{2 sin(}$$_{lπt}$$_{)}$$_{, for}$ s and t in [0 , 1] . The raw observations were generated following (1) and (2), with random s$_{ij}$ s uniformly distributed in [0 , 1] , { ε$_{ij}$ } and { ε ′ ij } independent N(0 , 0 . 01) , and m$_{i}$ = m ′ i = m ; for simplicity we took the grid { t$_{ij}$ } equal to { s$_{ij}$ } . The first series of simulations were designed to study estima tion error of the ˆ Θ s, both for clean and for outlier-contaminated data. We generated outliers by replacing [ εn ] of the pairs ( U$_{i}$ , V$_{i}$ ) by ( U ∗ $_{i}$, V ∗ $_{i}$) , with U ∗ i 1 = U$_{i}$$_{1}$ + 5 and U ∗ ij = U$_{ij}$ for j = 1 , and V ∗ i = W$_{i}$ . Note that the contaminated data ( U ∗ $_{i}$, V ∗ $_{i}$) follows model (8) with Θ$_{0}$ = O and high- leverage U ∗ $_{i}$s, so the effect of this type of contamination is an underestimation of θ$_{0}$$_{,}$$_{11}$ that tends to pull ˆ β ( s, t ) towards 0. The estimation of Θ$_{0}$ requires two steps: first, to estimate { U$_{i}$ } and { V$_{i}$ } from the raw data, and then to compute ˆ Θ from the ˆ U$_{i}$ s and the ˆ V$_{i}$ s. So we compared two procedures: a non-robust procedure, using reduced-rank Normal models (James et al., 2000) to estimate the component scores, followed by the ordinary least-squares regression estimator (9); and a robust procedure, using reduced-rank t -models (Gervini, 2009) to estimate the compo- nent scores, followed by the GMt regression estimator (10). For the robust procedure, we considered the two types of weights w ( ˆ U$_{i}$ ) discussed in Section 2.2, with trimming pro- portions α = . 10 and α = . 50 ; degrees of freedom ν = 1 and ν = 5 were used for the t -models. 9