4 one can assume that var( U$_{1}$ ) ≥ var( U$_{2}$ ) ≥ · · · > 0 and var( V$_{1}$ ) ≥ var( V$_{2}$ ) ≥ · · · > 0 .) This is the functional equivalent of the principal-component decomposition in multivariate analysis, so the φ$_{k}$ ( s ) s and ψ$_{l}$ ( t ) s are called “principal components”, and the U$_{k}$ s and V$_{l}$ s are called “component scores”. In principle p and q in (3) and (4) could be infinite, but since E( ‖ X − µ$_{X}$ ‖ $^{2}$) = ∑ p k $_{=1}$var( U$_{k}$ ) and E( ‖ Y − µ$_{Y}$ ‖ $^{2}$) = ∑ q l $_{=1}$var( V$_{l}$ ) are finite, the sequences { var( U$_{k}$ ) } and { var( V$_{l}$ ) } usually decrease to zero fast enough that for practical purposes p and q can be assumed to be finite. Methods for estimating the mean and the principal components of X ( s ) and Y ( t ) can be found in Ramsay and Silverman (2005), James et al. (2000), and Yao et al. (2005b). These methods are not resistant to outliers, though; outlier-resistant estimators of the mean and principal components have been proposed by Locantore et al. (1999), Cuevas et al. (2007), and Gervini (2008, 2009). We will use the method of Gervini (2009) to estimate the mean and the principal components in (3) and (4). This method is briefly reviewed in the Appendix. Now suppose that there is a functional linear relationship between X ( s ) and Y ( t ) : where α$_{0}$ ( t ) is the intercept, β$_{0}$ ( s, t ) the slope, and Z ( t ) the error term. We assume E { Z ( t ) } = 0 and cov { X ( s ) , Z ( t ) } = 0 for all s and t . (Note that the Z is not neces- sarily white noise; it is just the portion of Y that is not explained by X , and it is usually a smooth non-trivial process.) Since (5) implies that µ$_{Y}$ ( t ) = α$_{0}$ ( t ) + ∫ b a β$_{0}$ ( s, t ) µ$_{X}$ ( s ) ds , Y ( t ) = α$_{0}$ ( t ) + ∫ b a β$_{0}$ ( s, t ) X ( s ) ds + Z ( t ) , (5)