Dimension reduction for regression over a general class of functions

Quentin Paris (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Given an $\R^p\times\R$ valued random variable $(X,Y)$, we investigate a new and nonparametric dimension reduction approach for estimating the regression function $r(x)=\esp(Y\vert X=x)$ when $p$ is large. We assume given a class $\mathcal F$ of functions $\varphi:\Rp\ra\Rp$ such that there exists $\varphi\in\mathcal F$ with $$\esp(Y\vert \varphi(X))=\esp(Y\vert X).$$ In classical sufficient dimension reduction, one considers linear transformations of the predictor variable $X$ that preserve the conditional expectation. Extending this approach, the class $\mathcal F$ is considered here to be a general and possibly nonparametric class of functions. In this context, we introduce the {\it reduced dimension} $d_{\mathcal F}$ associated with $\mathcal F$, defined as the dimension of the lowest dimensional subspace of $\Rp$ spanned by the range of a function $\varphi\in\mathcal F$ satisfying the former equality. Then, we define an estimate $\hat r$ of $r$ and we prove that $\hat r$ achieves the optimal rate of convergence as if the predictor $X$ where $d_{\mathcal F}$-dimensional.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Quentin Paris. Dimension reduction for regression over a general class of functions. 2012. ⟨hal-00785643v1⟩