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## 2.2 Outlier-resistant functional regression 

As explained above, given the data ( x$_{1}$ , y$_{1}$ ) , . . . , ( x$_{n}$ , y$_{n}$ ) we use the reduced-rank t esti- mators of Gervini (2009) to obtain robust estimators of µ$_{X}$ , µ$_{Y}$ , { φ$_{k}$ } , { ψ$_{l}$ } , { U$_{ik}$ } and { V$_{il}$ } . By (7) and (8), the least-squares estimator of β$_{0}$ ( s, t ) would be φ ( s ) T ˆ Θ ψ ( t ) with 

<img src='content_image/315608.jpg'>

However, this estimator is not robust. Although the reduced-rank t estimators of µ$_{X}$ , µ$_{Y}$ , { φ$_{k}$ } and { ψ$_{l}$ } are robust, the component scores ˆ U$_{i}$ and ˆ V$_{i}$ are individual parameters that will be outliers if the corresponding curves X$_{i}$ ( s ) and Y$_{i}$ ( t ) are outliers. Therefore, the estimator of Θ$_{0}$ has to incorporate a mechanism to downweight outlying ˆ U$_{i}$ s and ˆ V$_{i}$ s. 

This can be accomplished, for instance, by a modification of t he t -type GM-estimators of He et al. (2000), that we will call GMt for short. Let 

<img src='content_image/315607.jpg'>

where ρ ( x ) = ( ν + q ) log (1 + x/ν ) . These are the maximum likelihood estimators of Θ$_{0}$ and Σ$_{0}$ when W in (8) follows a multivariate t distribution with mean zero and scatter matrix Σ$_{0}$ /w ( ˆ U$_{i}$ ) , although we do not actually assume that W follows this distribution; as in He et al. (2000), this is just the motivation behind defin ition (10). 

