A generalized distance for inference on functional data

It is well known that not all the inferential procedures adopted in the multivariate PCA can be traightforwardly extended to the functional case. More specifically, the inference on the mean is typically based on the Mahalanobis distance, which is in general undefined when data belongs to an infinite dimensional space. However, the common approach to consider few principal components is in contrast with some properties of the Mahalanobis distance and it may cause a loss of information. To address this issue, we propose a generalization of Mahalanobis distance for functional data, which is able to: (i) consider all the infinite components of data basis expansion and (ii) present features similar to the Mahalanobis distance. This new metric is adopted in an inferential context to construct tests on the mean of Gaussian processes.