Hotelling in wonderland

While Hotelling’s T2 statistic is traditionally defined as the Mahalanobis distance between the sample mean and the true mean induced by the inverse of the sample covariance matrix, we hereby propose an alternative definition which allows a unifying and coherent definition of Hotelling’s T2 statistic in any Hilbert space independently from its dimensionality and sample size. In details, we introduce the definition of random variables in Hilbert spaces, the concept of mean and covariance in such spaces and the relevant operators for formulating a proper definition of Hotelling’s T2 statistic relying on the concept of Bochner integral.