The author was partly supported by NSF grants DMS 0604396 and 1006281. The method proposed by Gervini (2009) to estimate the mean and the principal compo- nents of a stochastic process X works as follows. The mean function µ$_{X}$ and the princi- pal components { φ$_{k}$ } are modeled as spline functions; that is, given a set of spline basis functions b$_{1}$, . . . , b$_{N}$ , chosen by the user, it is assumed that µ$_{X}$ ( s ) = ∑ N l =1 ξ$_{l}$b$_{l}$ ( s ) and φ$_{k}$ ( s ) = ∑ N l =1 η$_{kl}$b$_{l}$ ( s ) . The observed vector x$_{i}$ can then be expressed as where B$_{i}$ = [ b$_{l}$ ( s$_{ij}$ )]$_{(}$$_{j,l}$$_{)}$ , H = [ η$_{1}$ , . . . , η$_{p}$ ] and Λ = diag( λ$_{1}$, . . . , λ$_{p}$ ) . Note that U$_{i}$ = Λ 1 / $^{2}$z$_{i}$ in this notation. By assuming ( z$_{i}$ , ε$_{i}$ ) has a standard multivariate t distribution, ro- bust maximum likelihood estimators of ξ , { η$_{k}$ } , { λ$_{k}$ } and σ are obtained. The estimators are computed via a standard EM algorithm. The optimal number of components p can be chosen via AIC or BIC criteria. See Gervini (2009) for details. In addition to parame- ter estimates, the EM algorithm yields predictors of the random effects z$_{i}$ , so one obtains ˆ U$_{i}$ = ˆ Λ 1 / $^{2}$ˆz$_{i}$ as a by-product. The estimators of µ$_{Y}$ , { ψ$_{k}$ } , and { V$_{i}$ } are obtained in a similar way from the sample y$_{1}$ , . . . , y$_{n}$ . 17 x$_{i}$ = B$_{i}$ ξ + B$_{i}$HΛ 1 / $^{2}$z$_{i}$ + σ ε$_{i}$ , Acknowledgement Appendix Reduced-rank t models