and (11) follows. Differentiating m with respect to Σ we obtain d m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = = ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}${− Σ − $^{1}$(d Σ ) Σ − $^{1}$} ( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) + tr { Σ − $^{1}$(d Σ ) } = − ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )tr { Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$(d Σ ) } + tr { Σ − $^{1}$(d Σ ) } = − ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )vec { Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$} $^{T}$vec(d Σ ) +vec( Σ − $^{1}$) $^{T}$vec(d Σ ) , ∇ vec( Σ $_{)}$m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = − ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )vec { Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$} + vec( Σ − $^{1}$) . ∂ ∂ Σ m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = − ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ ) Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − 1 + Σ − $^{1}$, 19 from which (12) follows. We will simplify the derivation of the asymptotic distribution of ˆ Θ by assuming that the true component scores ( U$_{i}$ , V$_{i}$ ) are used, instead of the estimated scores ( ˆ U$_{i}$ , ˆ V$_{i}$ ) , and by assuming that Σ$_{0}$ is fixed and known. In that case we can apply Theorem 5.23 of Van der Vaart (1998) directly, and obtain that √$_{n}$ { vec( ˆ Θ ) − vec( Θ$_{0}$ ) } is asymptotically so Again, this can be expressed in matrix form as