= 2 ρ $^{′′}$( e ) w $^{2}$( U )vec { U ( V − Θ $^{T}$U ) $^{T}$Σ − 1 0 } vec { U ( V − Θ $^{T}$U ) $^{T}$}T + ρ $^{′}$( e ) w ( U )( I$_{q}$ ⊗ UU $^{T}$) = 2 ρ $^{′′}$( e ) w $^{2}$( U )( Σ − 1 0 R ⊗ U )( R ⊗ U ) T + ρ $^{′}$( e ) w ( U )( I$_{q}$ ⊗ UU $^{T}$) , from which (15) follows. References Ash, R. B. and Gardner, M. F. (1975). Topics in Stochastic Processes . Probability and Mathematical Statistics (Vol. 27). New York: Academic Press. Chiou, J.-M., M¨uller, H.-G., and Wang, J.-L. (2004). Functional response models. Sta- tistica Sinica 14 , 675–693. Cuevas, A., Febrero, M., and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 , 481–496. Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press. Fraiman, R., and Muniz, G. (2001). Trimmed means for functional data. Test 10 , 419– 440. Gervini, D. (2008). Robust functional estimation using the median and spherical princi- pal components. Biometrika 95 , 587–600. 21