Hotelling's $T^2$ in separable Hilbert spaces

Hotelling's $T^2$ in separable Hilbert spaces
Monday 20th February 2017
Pini, A.; Stamm, A.; Vantini, S.
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We address the problem of finite-sample null hypothesis significance testing on the mean element of a random variable that takes value in a generic separable Hilbert space. For this purpose, we propose a (re)definition of Hotelling's $T^2$ statistic that naturally expands to any separable Hilbert space that we further embed within a permutation inferential approach. In detail, we present a unified framework for making inference on the mean element of Hilbert populations based on Hotelling's $T^2$ statistic, using a permutation-based testing procedure of which we prove finite-sample exactness and consistency; we showcase the explicit form of Hotelling's $T^2$ statistic in the case of some famous spaces used in functional data analysis (i.e., Sobolev and Bayes spaces); we propose simulations and a case study that demonstrate the importance of the space into which one decides to embed the data; we provide an implementation of the proposed tools in the R package fdahotelling