Statistical inference in compound functional models

TL;DR

This paper introduces a comprehensive nonparametric regression framework called the compound model, characterizing its parameters and deriving optimal convergence rates for estimators, adaptable to various sparse and additive structures.

Contribution

It provides the first non-asymptotic minimax convergence rates for the compound model, encompassing sparse additive and high-dimensional regression, with adaptive estimation strategies.

Findings

Derived non-asymptotic minimax rates of convergence.

Established adaptive estimation procedures.

Unified analysis for various sparse and additive models.

Abstract

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.