Functional nonparametric statistics: a double infinite dimensional framework

Functional aspects are more and more frequent and varied in modern Statistics so much so that the designation of Functional Statistics had emerged recently (see [46]). A symbolical example of this new field of Statistics concerns the problems of nonparametric estimation in presence of functional data, which are doubly infinite dimensional problems since the functional aspects appears twice: in the nature of the observed data and in the object to be estimated which derives from the statistical model. We will give a comprehensive introduction to nonparametric modelling for functional variables, together with the presentation of the state of art in this recent field of Statistics. We will present several different situations (such as regression, conditional cumulative distribution, conditional density, conditional mode and quantiles, density estimation, time series prediction, supervised curves classification,...), but we will pay special attention to the regression problem. We present one selected recent theoretical result in regression in order to explain the double dimensional effects. The description of the state of art, as well in regression as for other mentionned functional problems, will be articulated through this selected asymptotic result. At the end, several open problems and suggestions for future researches are discussed.