Asymptotic equivalence for nonparametric regression with multivariate and random design

By Markus Reiß. 2006

112 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1843fbf75c392cfa9f71810b4ae6e659c8c82464

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Citations in the corpus

Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “In common cases where λ represents a uniform distribution on some bounded domain in Euclidean space, a deterministic design regression model with ‘equally spaced’ design Xi = xi can be seen to be statistically equivalent to (4), see [51], and our analysis thus also informs such measurement setups.”
NONPARAMETRIC ESTIMATION OF COMPOSITE FUNCTIONS (2009, methodology): “ion and classiﬁcation. Note that our theorems can be directly transposed to the Gaussian nonparametric regression model with ﬁxed equidistant design using the equivalence of experiments argument (cf. [24]). Note also that results similar to ours have been recently obtained for the problem of testing”
Tight conditions for consistency of variable selection in the context of high dimensionality (2011, methodology): “To convey the main ideas without taking care of some technical details, we start by focusing our attention on the Gaussian white noise model, that was proved to be asymptotically equivalent to the model of regression [ 6, 35 ] , as well as to other nonparametric models [ 7, 13 ] .”
Asymptotic equivalence for inference on the volatility from noisy observations (2011, methodology): “Since by the classical result of Brown and Low (1996) or by Reiß (2008) the Gaussian regression is equivalent to the corresponding white noise experiment [note that log(σ2(•) + h−20 π2) is also αHölder continuous], we have already derived an important and far-reaching result.”
Optimal inference in a class of regression models (2015, unknown): “model, which can be obtained as a limiting model for nonparametric density estimation (see Nussbaum, 1996) as well as nonparametric regression with ﬁxed or random regressors (see Brown and Low, 1996; Reiß, 2008). These white noise equivalence results imply that our ﬁnite-sample results translate to ”