Evaluating the complexity of some families of functional data

In this paper we study the complexity of functional data set by means of a two steps approach. The first step considers a new graphical tool for assessing to which family the data belong: the main aim is to detect whether a sample comes from a monomial or an exponential family. This first tool is based on a nonparametric kNN estimation of Small Ball Probability. Once the family is specified, the second step consists in evaluating the extent of complexity by estimating some specific indexes related to the assigned family. It turns out that the developed methodology is fully free from assumptions on model, distribution as well as dominating measure. This large flexibility ensures the wide applicability of the methodology. Computational issues are carried out by means of simulations and finally the method is applied to analyze some financial real curves dataset.