Estimation locale linéaire de la fonction de régression pour des variables hilbertiennes

AbstractIn this paper, we introduce a new nonparametric estimation of the regression function when both the response and the explanatory variables are of the functional kind. First, we construct a local linear estimator of this regression operator, then we state its rate for the uniform almost complete convergence. This latter is expressed as a function of the small ball probability of the predictor and as a function of the entropy of the set on which the uniformity is obtained

Choosing the most relevant level sets for depicting a sample of densities

The final publication is available at link.springer.comWhen exploring a sample composed with a set of bivariate density functions, the question of the visualisation of the data has to front with the choice of the relevant level set(s). The approach proposed in this paper consists in defining the optimal level set(s) as being the one(s) allowing for the best reconstitution of the whole density. A fully data-driven procedure is developed in order to estimate the link between the level set(s) and their corresponding density, to construct optimal level set(s) and to choose automatically the number of relevant level set(s). The method is based on recent advances in functional data analysis when both response and predictors are functional. After a wide description of the methodology, finite sample studies are presented (including both real and simulated data) while theoretical studies are reported to a final appendix.Peer ReviewedPostprint (author's final draft

A goodness-of-fit test for the functional linear model with functional response

The Functional Linear Model with Functional Response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this paper, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cram\'er-von Mises norm over a doubly-projected empirical process which, using geometrical arguments, yields an easy-to-compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use of convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real datasets illustrate the application of the new test.Comment: 24 pages, 2 figures, 10 tables. Suplementary material: 2 pages, 1 figur