GRID: A variable selection and structure discovery method for high dimensional nonparametric regression

Abstract

We consider nonparametric regression in high dimensions where only a relatively small subset of a large number of variables are relevant and may have nonlinear effects on the response. We develop methods for variable selection, structure discovery and estimation of the true low-dimensional regression function, allowing any degree of interactions among the relevant variables that need not be specified a-priori. The proposed method, called the GRID, combines empirical likelihood based marginal testing with the local linear estimation machinery in a novel way to select the relevant variables. Further, it provides a simple graphical tool for identifying the low dimensional nonlinear structure of the regression function. Theoretical results establish consistency of variable selection and structure discovery, and also Oracle risk property of the GRID estimator of the regression function, allowing the dimension $d$ of the covariates to grow with the sample size $n$ at the rate $d=O(n^{a})$ for any $a\in(0,\infty)$ and the number of relevant covariates $r$ to grow at a rate $r=O(n^{\gamma})$ for some $\gamma\in(0,1)$ under some regularity conditions that, in particular, require finiteness of certain absolute moments of the error variables depending on $a$. Finite sample properties of the GRID are investigated in a moderately large simulation study.