B0982
Title: Inference on the means of functional data based on a generalized Mahalanobis distance
Authors: Andrea Ghiglietti - Universita degli Studi di Milano (Italy) [presenting]
Anna Maria Paganoni - MOX-Politecnico di Milano (Italy)
Francesca Ieva - Politecnico di Milano (Italy)
Abstract: The problem of testing about the means of two independent samples of functional data is considered. Several inferential tools used in the classical multivariate analysis are not useful in the infinite dimensional framework of the functional data analysis. For instance, all the procedures based on the popular Mahalanobis distance are not allowed since this metrics is not well-defined in $L^2$. Then, the problem is typically solved by reducing the inference on a finite dimensional space given by few principal components, but this approach is not always satisfactory, since it may lead to a loss of information and some of its properties are in contrast with the Mahalanobis approach. For this reason, we propose a distance that generalizes the Mahalanobis metrics in infinite dimensional Hilbert space of square integrable functions defined on a compact interval, without any truncation on the number of components considered in the distance. We present a convergence result and a central limit theorem for the generalized distance between the means of two independent samples of curves without specifying the probability distribution of the processes generating the data. These asymptotic results allow us to construct critical regions to test the means of two-samples and to compute analytically the relative power. Finally, these inferential procedures are applied to a case study concerning samples of ECG curves.
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