Asymptotic equivalence for nonparametric additive regression

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~$Γ$. 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 $Γ$ splits into a multiplication operator and an asymptotically negligible Hilbert-Schmidt operator.

💡 Research Summary

The paper establishes a rigorous Le Cam asymptotic equivalence between the classical non‑parametric additive regression model and a specially constructed Gaussian white‑noise experiment. In the regression setting one observes i.i.d. pairs ((X_j,Y_j)) with
(Y_j = g(X_j) + \varepsilon_j,)
where the regression function has an additive structure
(g(x)=\sum_{\ell=1}^{d} g_\ell(x_\ell))
and the errors are independent (N(0,\sigma^2)). The covariates (X_j) are drawn from a known density (p_X) on (