β ( t ) = q ∑ k =1 V$_{k}$ψ$_{k}$ ( t ) , (8) where { φ$_{k}$ ( t ) } and { ψ$_{k}$ ( t ) } are orthonormal functions in L $^{2}$( I ) , the U$_{k}$ s are uncorrelated with E( U$_{k}$ ) = 0 and var( U$_{k}$ ) = γ$_{k}$ , and the V$_{k}$ s are uncorrelated with E( V$_{k}$ ) = 0 and var( V$_{k}$ ) = λ$_{k}$ . Without loss of generality we assume γ$_{1}$ ≥ · · · ≥ γ$_{p}$ > 0 and λ 1 ≥ · · · ≥ λ$_{q}$ > 0 . From (6), (7) and (8) it follows that the total variance of z$_{ij}$ ( t ) , defined as E( ‖ z$_{ij}$ − µ ‖ $^{2}$) with ‖·‖ the usual L $^{2}$-norm, can be decomposed as E( ‖ α ‖ $^{2}$) + E( ‖ β ‖ $^{2}$) , where E( ‖ α ‖ $^{2}$) = ∑ p k =1 γ$_{k}$ is the main-factor variance and E( ‖ β ‖ $^{2}$) = ∑ q k =1 λ$_{k}$ is the residual- factor variance. The ratio h$_{z}$ = ∑ p k =1 γ$_{k}$ ∑ p k =1 γ$_{k}$ + ∑ q k =1 λ$_{k}$ (9) is then the proportion of amplitude variability explained by the main factor. In Section 4 we will derive asymptotic confidence intervals for h$_{z}$ . The mean function µ ( t ) and the Karhunen–Lo`eve components { φ$_{k}$ ( t ) } and { ψ$_{k}$ ( t ) } are functional parameters that must be estimated from the data, using for instance semipara- metric spline models. Let b ( t ) = ( b 1 ( t ) , . . . , b$_{s}$ ( t )) T be a spline basis in L $^{2}$( I ) (for sim- plicity we will use the same spline basis for all functional parameters, but this is not strictly necessary); then we assume µ ( t ) = b ( t ) $^{T}$m , φ$_{k}$ ( t ) = b ( t ) $^{T}$c$_{k}$ , and ψ$_{k}$ ( t ) = b ( t ) $^{T}$d$_{k}$ , for parameters m , c$_{k}$ and d$_{k}$ in R $^{s}$. Let C = [ c$_{1}$ , . . . , c$_{p}$ ] ∈ R s × $^{p}$, D = [ d$_{1}$ , . . . , d$_{q}$ ] ∈ R s × q and J = ∫ b a b ( t ) b ( t ) $^{T}$dt ∈ R s × $^{s}$. The orthogonality conditions on the φ$_{k}$ s and the ψ$_{k}$ s translate into the conditions C $^{T}$JC = I$_{p}$ and D $^{T}$JD = I$_{q}$ for C and D . Regarding the U$_{k}$ s and V$_{k}$ s in (7) and (8), we assume that U = ( U 1 , . . . , U$_{p}$ ) T follows a multivariate N ( 0 , Γ ) distribution with Γ = diag( γ$_{1}$, . . . , γ$_{p}$ ) and that V = ( V 1 , . . . , V$_{q}$ ) T follows a multivariate N ( 0 , Λ ) distribution with Λ = diag( λ 1 , . . . , λ$_{q}$ ) . To summarize, the warped process (6) is parameterized by m , C , D , Γ and Λ . For the warping functions w$_{ij}$ ( t ) we cannot simply assume an additive model like (6) 8