Reiß, Markus, Asymptotic Equivalence for Nonparametric Regression with Multivariate and Random Design, Annals of statistics , 36(4), 2008, pp. 1957-1982
We show that nonparametric regression is asymptotically equivalent, in Le Cam's sense, to a sequence of Gaussian white noise experiments as the number of observations tends to infinity. We propose a general constructive framework, based on approximation spaces, which allows asymptotic equivalence to be achieved, even in the cases of multivariate and random design.