Title: Inference in separable Hilbert spaces using Hotelling's T2
Authors: Aymeric Stamm - CNRS (French National Center for Scientific Research) (France) [presenting]
Alessia Pini - Università Cattolica del Sacro Cuore (Italy)
Simone Vantini - Politecnico di Milano (Italy)
Abstract: Hotelling's T2 is introduced in multivariate data analysis courses for parametric hypothesis testing on the mean vector of multivariate Gaussian distributions. In details, given a sample of $n$ i.i.d. random variables following a $p$-variate Gaussian distribution, under the null hypothesis that the mean vector is equal to some fixed value, a properly scaled version of Hotelling's T2 statistic follows a Fisher distribution, provided that $n > p$. When either the data does not follow a Gaussian distribution or its dimension exceeds the sample size, this result does not hold anymore, which has led the statistical community to move away from Hotelling's T2 and introduce new statistics along with non-parametric approaches to solve one- and two-sample testing problems. Situations like this naturally arise from high-dimensional data (i.e. when $p > n$), from functional data (where $p$ is actually infinite) or, more generally, from object data which belong to Hilbert spaces or even metric spaces. We will show that Hotelling's T2 is in fact well defined in generic Hilbert spaces, and we will provide a practical way of computing its value in separable Hilbert spaces. Next, we will provide an exact permutation testing procedure based on Hotelling's T2 statistic for solving the one- and two-sample problems in separable Hilbert spaces. We will show simulations and a case study on Aneurysm data as basis for discussing the performances of the approach.