Statistical inference in compound functional models

Arnak S. Dalalyan (1, 2, 3) , Yuri Ingster (4, 5) , Alexandre Tsybakov (1, 6)

1 CREST - Centre de Recherche en Économie et Statistique
2 imagine [Marne-la-Vallée]
3 LIGM - Laboratoire d'Informatique Gaspard-Monge
4 St. Petersburg State Electrotechnical University
5 LETI - Saint Petersburg Electrotechnical University
6 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider a general nonparametric regression model called the compound model. It includes, as special cases, sparse additive regression and nonparametric (or linear) regression with many covariates but possibly a small number of relevant covariates. The compound model is characterized by three main parameters: the structure parameter describing the "macroscopic" form of the compound function, the "microscopic" sparsity parameter indicating the maximal number of relevant covariates in each component and the usual smoothness parameter corresponding to the complexity of the members of the compound. We find non-asymptotic minimax rate of convergence of estimators in such a model as a function of these three parameters. We also show that this rate can be attained in an adaptive way.

Mots clés: Compound functional model; minimax estimation; sparse additive structure; dimension reduction; structure adaptation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Arnak S. Dalalyan, Yuri Ingster, Alexandre Tsybakov. Statistical inference in compound functional models. 2012. ⟨hal-00725663v2⟩