and the amplitude process z$_{ij}$ ( t ) . This can be done by assuming that ( u$_{i}$ , η$_{i}$ ) and ( v$_{ij}$ , ξ$_{ij}$ ) have joint Normal distributions, for instance. The only change in (12) would be that f ( u$_{i}$ ) f ( η$_{i}$ ) is replaced by the joint density f ( u$_{i}$ , η$_{i}$ ) and f ( v$_{ij}$ ) f ( ξ$_{ij}$ ) by f ( v$_{ij}$ , ξ$_{ij}$ ) . From the computational point of view this does not have a big impact, because the EM algorithm can be easily modified to accommodate this (a Matlab implemen tation is available as sup- plementary material). But from a statistical point of view the results may be harder to interpret, and the extra rp + rq covariance parameters that need to be estimated may affect the precision of the rest of the estimators if the sample size is not very large. Finally, we note that the assumption of normality of the random effects is mostly a working assumption to derive estimators. It is usually the case that properties like consis- tency and asymptotic normality of maximum likelihood estimators hold for broader fam- ilies of distributions than the one they were derived for. But such a thorough asymptotic analysis is beyond the scope of this paper. Instead, in Section 5 we will study by simula- tion the robustness of the estimators to at least some mild departures from normality. On the other hand, if robustness to outliers is desired, this may be attained by substituting the Normal distributions by multivariate t distributions, as in Gervini (2009); the Normal EM algorithm is easy to adapt for multivariate t distributions. It is usually of interest in applications to determine if the main-factor variance is signif- icantly different from zero or not. To this end, we derive in this section the asymptotic distributions of the maximum likelihood estimators and the variance ratios (9) and (11), which can then be used to construct asymptotic confidence int ervals and tests for h$_{z}$ and h$_{w}$ . For simplicity, we assume that (i) the true functional parameters µ ( t ) , { φ$_{k}$ ( t ) } and { ψ$_{k}$ ( t ) } belong to the spline space used for estimation, which is fixed , and (ii) the y$_{i}$$_{·}$ s are 11 4 Asymptotics and inference