Recent technological advances have eased the collection of big amounts of data in many
research fields. In this scenario, a useful statistical technique is density estimation which
represents an important source of information. One dimensional density functions represent a
special case of functional data subject to the constraints to be non-negative and with a constant
integral equal to one. Because of these constraints, densities functions do not form a vector
space and a naive application of functional data analysis (FDA) methods may lead to nonvalid estimates. To address this issue, two main strategies can be found in the literature. In the
first, the probability density functions (pdfs) are mapped into a linear functional space through
a suitably chosen transformation. Established methods for Hilbert space valued data can be
applied to the transformed functions and the results are moved back into the density space by
means of the inverse transformation. In the second strategy, probability density functions are
treated as an infinite dimensional compositional data since they are part of some whole which
only carry relative information. In this work, by means of a suitable transformation, densities
are embedded in the Hilbert space of square integrable functions where standard FDA
methodologies can be applied.
28 Ottobre 2022
Stefano Antonio Gattone
University G. d'Annunzio of Chieti-Pescara