Robust functional data analysis: From sparse to dense designs

We propose a new perspective to conduct robust functional data analysis of discretely observed functional data ranging from sparse to dense sampling designs. This analysis caters to processes with various distributions, including heavy-tailed, skewed, or contaminated distributions. We study the robust functional mean (M-location) and introduce a robust dimension reduction method via principal component analysis. Theoretical outcomes for the robust functional mean and eigenfunction estimates, derived from pooled discretely observed data, are elucidated, matching their non-robust counterparts. The established convergence rates for these estimated eigenfunctions, with indices increasing with sample size, pave the way for further modeling and analysis.