Quantiles for finite and infinite dimensional data

A new projection-based definition of quantiles in a multivariate setting is proposed. This approach extends in a natural way to infinite-dimensional Hilbert spaces. The directional quantiles we define are shown to satisfy desirable properties of equivariance and, from an interpretation point of view, the resulting quantile contours provide valuable information when plotting them. Sample quantiles estimating the corresponding population quantiles are defined and consistency results are obtained. The new concept of principal quantile directions, closely related in some situations to principal component analysis, is found specially attractive for reducing the dimensionality and visualizing important features of functional data. Asymptotic properties of the empirical version of principal quantile directions are also obtained. Based on these ideas, a simple definition of robust principal components for finite and infinite-dimensional spaces is also proposed. The presented methodology is illustrated with examples throughout the paper.