content_image/301546.jpg content_image/301545.jpg content_image/301544.jpg content_image/301543.jpg
## GMt estimating equations and asymptotics 

The estimators ˆ Θ and ˆ Σ defined by (10) are M-type estimators (Van der Vaart, 1998, ch. 5), since they minimize a function of the form M ( Θ , Σ ) = 1 n ∑ n i $_{=1}$m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) . Specifically, 

<img src='content_image/301546.jpg'>

Then ˆ Θ and ˆ Σ solve the equations ∂ ∂ $_{Θ}$M ( ˆ Θ , ˆ Σ ) = O and ∂ ∂ $_{Σ}$M ( ˆ Θ , ˆ Σ ) = O . To com- pute matrix derivatives we use the method of differentials (Magnus and Neudecker, 1999). Differentiating with respect to Θ we obtain 

<img src='content_image/301545.jpg'>

where e$_{i}$ = w ( ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) . Then 

<img src='content_image/301544.jpg'>

which can be rearranged in matrix form as 

<img src='content_image/301543.jpg'>

