Asymptotic equivalence for nonparametric additive regression

TL;DR

This paper establishes that nonparametric additive regression models are asymptotically equivalent to certain Gaussian white noise models, facilitating analysis and inference in high-dimensional settings with increasing components.

Contribution

It proves asymptotic equivalence between additive regression and Gaussian white noise models, including cases with increasing additive components and specific covariate independence structures.

Findings

Asymptotic equivalence holds for increasing number of additive components.

Decomposition into independent processes for pairwise independent covariates.

Approximation results in semiparametric settings with operator splitting.

Abstract

We prove asymptotic equivalence of nonparametric additive regression and an appropriate Gaussian white noise experiment in which a multidimensional shifted Wiener process is observed, whose dimension equals the number of additive components. The shift depends on the additive components of the regression function and solely the one- and two-dimensional marginal distributions of the covariates via an explicitly specified bounded but non-compact linear operator~$\Gamma$. The number of additive components $d$ is allowed to increase moderately with respect to the sample size. In the special case of pairwise independent components of the covariates, the white noise model decomposes into $d$ independent univariate processes. Moreover, we study approximation in some semiparametric setting where $\Gamma$ splits into a multiplication operator and an asymptotically negligible Hilbert-Schmidt operator.