|Abstract:|| The field of statistics is at the cusp of a revolution in the way data is collected by measuring instruments. Massive information is retrieved in real-time and/or spatially-referenced, hence producing new kind of data: functional data. Statistical inference for functional data is particularly challenging as it is an extreme case of high-dimensional data for which, no matter how large the sample is, information will always be insufficient to fully characterize the underlying model.
In detail, after a historical excursus over the test statistics introduced for approaching the problem of testing the mean, we provide a generalization of Hotelling s $T^2$ on any functional Hilbert space, naturally dubbed functional Hotelling s $T^2$. We discuss a nonparametric permutational framework that enables statistical testing for the mean function of a population as well as for the difference between the mean functions of two populations. Within this framework, we show how a number of state-of-the-art test statistics can be seen as approximations of functional $T^2$ statistic hereby proposed.|