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b$_{l}$ { t ∗ $_{ijk}$( θ$_{ij}$ ) } . Then 

<img src='content_image/300588.jpg'>

and the y$_{ij}$ s are conditionally independent given ( u$_{i}$ , v$_{ij}$ , η$_{i}$ , ξ$_{ij}$ ) . If y$_{i}$$_{·}$ = ( y$_{i}$$_{1}$ , . . . , y$_{iJ}$$_{i}$ ) , we have 

<img src='content_image/300589.jpg'>

with g ( u$_{i}$ , η$_{i}$ ) = ∏ J$_{i}$ j =1 ∫∫ f ( y$_{ij}$ | u$_{i}$ , v$_{ij}$ , η$_{i}$ , ξ$_{ij}$ ) f ( v$_{ij}$ ) f ( ξ$_{ij}$ )d v$_{ij}$ d ξ$_{ij}$ , and the log-likelihood function is ℓ = ∑ I i $_{=1}$log f ( y$_{i}$$_{·}$ ) . The maximum likelihood estimators are ( ˆ m , ˆ C , ˆ D , ˆ Λ , ˆ Γ , ˆ Σ , ˆ Ω , ˆ σ $^{2}$) = arg max ℓ . We compute them via the EM algorithm. The implementation of the EM algorithm presents certain complications arising from the orthogonality restric- tions on C and D , which are discussed in detail in Web Appendix A. 

In the rest of the paper we will use the estimators as presented above, but to conclude the section we discuss a few possible generalizations. First, it is possible to use other fami- lies of warping functions, such as B-splines with monotone increasing coefficients (Brum- back and Lindstrom, 2004; Telesca and Inoue, 2008) or smooth monotone transformations (Ramsay and Li, 1998). The problem is that the spline coeffici ents for these families can- not be directly related to features of the sample curves in the way interpolating Hermite spline coefficients can; therefore, one may have to use a rela tively large number of knots placed at somewhat arbitrary locations (equally spaced, for example). This may result in a warping family that is too flexible and lead to overwarping ( i.e., produce warping func- tions with flat parts that are close to singular). To prevent t his, the warping variability must somehow be penalized. This can be done by adding a penalty term to the log-likelihood function and minimize ℓ$_{λ}$ = ∑ I i $_{=1}$log f ( y$_{i}$$_{·}$ ) − λ tr( Σ + Ω ) , where λ ≥ 0 is a penalty parameter chosen by the user. 

Second, it is possible to incorporate correlations between the warping process w$_{ij}$ ( t ) 

