A Flexibly Conditional Screening Approach via a Nonparametric Quantile Partial Correlation

Abstract

Considering the influence of conditional variables is crucial to statistical modeling, ignoring this may lead to misleading results. Recently, Ma, Li and Tsai proposed the quantile partial correlation (QPC)-based screening approach that takes into account conditional variables for ultrahigh dimensional data. In this paper, we propose a nonparametric version of quantile partial correlation (NQPC), which is able to describe the influence of conditional variables on other relevant variables more flexibly and precisely. Specifically, the NQPC firstly removes the effect of conditional variables via fitting two nonparametric additive models, which differs from the conventional partial correlation that fits two parametric models, and secondly computes the QPC of the resulting residuals as NQPC. This measure is very useful in the situation where the conditional variables are highly nonlinearly correlated with both the predictors and response. Then, we employ this NQPC as the screening utility to do variable screening. A variable screening procedure based on NPQC (NQPC-SIS) is proposed. Theoretically, we prove that the NQPC-SIS enjoys the sure screening property that, with probability going to one, the selected subset can recruit all the truly important predictors under mild conditions. Finally, extensive simulations and an empirical application are carried out to demonstrate the usefulness of our proposal.