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Asymptotic equivalence for nonparametric regression with multivariate and random design. (English) Zbl 1142.62023
Summary: We show that nonparametric regression is asymptotically equivalent, in Le Cam’s sense [L. Le Cam and G. Lo Yang, Asymptotics in statistics. Some basic concepts. 2nd ed. (2000; Zbl 0952.62002)], 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 designs.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62B15 Theory of statistical experiments
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