A theory of nonparametric regression in the presence of complex nuisance components

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In this paper, we consider the nonparametric random regression model Y=f₁(X₁)+f₂(X₂)+ε and address the problem of estimating the function f₁. The term f₂(X₂) is regarded as a nuisance term which can be considerably more complex than f₁(X₁). Under minimal assumptions, we prove several nonasymptotic L²(Pˣ)-risk bounds for our estimators of f₁. Our approach is geometric and based on considerations in Hilbert spaces. It shows that the performance of our estimators is closely related to geometric quantities, such as minimal angles and Hilbert-Schmidt norms. Our results establish new conditions under which the estimators of f₁ have up to first order the same sharp upper bound as the corresponding estimators of f₁ in the model Y=f₁(X₁)+ε. As an example we apply the results to an additive model in which the number of components is very large or in which the nuisance components are considerably less smooth than f₁. In particular, the results apply to an asymptotic scenario in which the number of components is allowed to increase with the sample size.