Nonparametric estimation in the presence of complex nuisance components

Martin Wahl (Universität Mannheim)
Wednesday, February 4, 2015 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider the nonparametric random regression model $Y=f_1(X_1)+f_2(X_2)+\epsilon$ and address the problem of estimating the function $f_1$. The term $f_2(X_2)$ is regarded as a nuisance term which can be considerably more complex than $f_1(X_1)$. Under minimal assumptions, we prove several nonasymptotic risk bounds for our estimators of $f_1$. 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 $f_1$ have up to first order the same sharp upper bound as the corresponding estimators of $f_1$ in the model $Y=f_1(X_1)+\epsilon$. As an example we apply the results to an additive model in which the nuisance components are considerably less smooth than $f_1$ or in which the number of components increases with the sample size. If time permits, we also present an extension of the theory to high-dimensional but sparse additive models.