Hotelling meets Hilbert: inference on the mean in functional Hilbert spaces

The talk will focus on 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 will present a definition of Hotelling's T-2 statistic that naturally expands to any separable Hilbert space. In detail, after having recalled the notion of Gelfand-Pettis integral in separable Hilbert spaces and introduced the definition of random variables in Hilbert spaces, and the derived concepts of mean and covariance in such spaces, we will 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. We will then present the theoretical properties of the procedure (i.e., finite-sample exactness and consistency) and show the explicit form of Hotelling's T-2 statistic in the case of some famous spaces used in functional data analysis like Sobolev and Bayes spaces. We will finally demonstrate the importance of the space into which one decides to embed the data by means of simulations and a case study.