# Probability and Statistics Seminar

### Robust estimators in functional principal components

When dealing with multivariate data, like classical PCA, robust PCA searches for directions with maximal dispersion of the data projected on it. Instead of using the variance as a measure of dispersion, a robust scale estimator s_n may be used in the maximization problem. This approach was first in Li and Chen (1985) while a maximization algorithm was proposed in Croux and Ruiz-Gazen (1996) and their influence function was derived by Croux and Ruiz-Gazen (2005). Recently, their asymptotic distribution was studied in Cui et al. (2003).

Let $X\left(t\right)$ be a stochastic process with continuous trajectories and finite second moment, defined on a finite interval. We will denote by $\Gamma \left(t,s\right)=cov\left(X\left(t\right),X\left(s\right)\right)$ its covariance function and by ${\varphi }_{j}$ and ${\lambda }_{j}$ the eigenfunctions and the eigenvalues of the covariance operator with ${\lambda }_{j}$ in the decreasing order. Dauxois et al. (1982) derived the asymptotic properties of non-smooth principal components of functional data obtained by considering the eigenfunctions of the sample covariance operator. On the other hand, Silverman (1996) and Ramsay and Silverman (1997), introduced smooth principal components for functional data, based on roughness penalty methods while Boente and Fraiman (2000) considered a kernel-based approach. More recent work, dealing with estimation of the principal components of the covariance function, includes Gervini (2006), Hall and Hosseini-Nasab (2006), Hall et al. (2006) and Yao and Lee (2006). Up to our knowledge, the first attempt to provide estimators of the principal components less sensitive to anomalous observations was done by Locantore et al. (1999) who considered the coefficients of a basis expansion. Besides, Gervini (2008) studied a fully functional approach to robust estimation of the principal components by considering a functional version of the spherical principal components defined in Locantore et al. (1999). On the other hand, Hyndman and Ullah (2007) provide a method combining a robust projection-pursuit approach and a smoothing and weighting step to forecast age-specific mortality and fertility rates observed over time.

In this talk, we introduce robust estimators of the principal components and we obtain their consistency under mild conditions. Our approach combines robust projection-pursuit with different smoothing methods.