Focusing on Structural Asumptions in Regression on Functional Variable
Laurent Delsol
MAPMO, Université d'Orléans, France

Functional statistics aims to provide relevant tools to deal with samples of curves (or more generally infinite dimensional variables). This field of modern statistics presents theoretical challenges and opens a wide scope of potential real world applications. Various well known multivariate models and methods have been extended to take into account the infinite dimension of the variables. A lot of consideration has been given to the study of the regression of a real-valued response on a functional variable. Many estimation procedures, often based on structural assumptions on the regression operator, have been considered. This work concerns an other type of issue: the construction of structural testing procedures in such regression models. Structural tests may be relevant by themselves to check the validity of some a priori model or some theoretical assumptions. They are also relevant complementary tools to estimation methods to check the validity of some structural assumptions used to construct the estimator or test if some heuristic hypothesis coming from an estimation of the regression operator holds. However, the literature devoted to such tests is reduced to a small number of papers. A theoretical background has been recently proposed to consider a wide scope of structural tests. The aim of this talk is to describe the practical use of some of these tests, discuss some recent improvements and talk about interesting prospects. The test statistic is computed from an estimator specific to the null (structural) hypothesis and uses recent advances in kernel smoothing for functional data. Then, several bootstrap methods are proposed to compute the threshold value. Simulation studies and applications are finally presented to illustrate the interest of the proposed testing procedures.

Keywords: Functional data; Structural test; Bootstrap; Regression

Biography: Laurent DELSOL made his PhD with Frédéric Ferraty and Philippe Vieu at the Mathematical Institute of Toulouse (University Paul Sabatier, Toulouse, France). He spent one year at the Statistical Institute of the Catholic University of Louvain-la-Neuve (Belgium) for a postdoctoral position and is now “Maître de Conférence” at MAPMO laboratory of Orleans University (France).