Nonparametric Model Checking and Variable Selection

TL;DR

This paper introduces a nonparametric ANOVA-type test for assessing the influence of individual covariates on a regression function, along with a variable selection method based on multiple testing, demonstrating superior performance in simulations and real data analysis.

Contribution

It develops a new nonparametric test for variable influence in regression models and proposes a variable selection method that outperforms existing procedures.

Findings

Test performs well under sparse models and extends to higher dimensions with dimension reduction.

Proposed variable selection method is competitive with established procedures.

Real data analysis validates the effectiveness of the methods.

Abstract

Let X be a d dimensional vector of covariates and Y be the response variable. Under the nonparametric model Y = m(X) + {\sigma}(X) \in we develop an ANOVA-type test for the null hypothesis that a particular coordinate of X has no influence on the regression function. The asymptotic distribution of the test statistic, using residuals based on Nadaraya-Watson type kernel estimator and d \leq 4, is established under the null hypothesis and local alternatives. Simulations suggest that under a sparse model, the applicability of the test extends to arbitrary d through sufficient dimension reduction. Using p-values from this test, a variable selection method based on multiple testing ideas is proposed. The proposed test outperforms existing procedures, while additional simulations reveal that the proposed variable selection method performs competitively against well established procedures. A real data set is analyzed.