Copula-based Partial Correlation Screening: a Joint and Robust Approach

Abstract

Screening for ultrahigh dimensional features may encounter complicated issues such as outlying observations, heterogeneous or heavy-tailed distribution, multi-collinearity and confounding effects. Standard correlation-based marginal screening methods may be a weak solution to these issues. We contribute a novel robust joint screener to safeguard against outliers and distribution mis-specification for both the response variable and the covariates, and to account for external variables at the screening step. Specifically, we introduce a copula-based partial correlation (CPC) screener. We show that the empirical process of the estimated CPC converges weakly to a Gaussian process and establish the sure screening property for CPC screener under very mild technical conditions, where we need not require any moment condition, weaker than existing alternatives in the literature. Moreover, our approach allows for a diverging number of conditional variables from the theoretical point of view. Extensive simulation studies 1 and two data applications are included to illustrate our proposal.

A note of feature screening via a rank‐based coefficient of correlation

“…In addition, (11) indicates that the proposed method is able to handle the nonpolynomial (NP) dimensionality with $$p = o\lbrace \exp (n^{1-2\kappa })\rbrace$$, which is parallel with other feature screening methods, such as distance correlation (Li et al., 2012b) or the rank concordance–based estimating equation (Ma et al., 2017). On the other hand, different from other feature screening methods that may require additional assumptions to establish theoretical results, such as the uniformly subexponential tail probability (e.g., Li et al., 2012b), specific requirement for the mean response (e.g., Ma et al., 2017), or boundness of density functions of Y and $$X_{(k)}$$ (e.g., Wu & Yin, 2015; Xia & Li, 2021), the proposed method requires fewer conditions to derive the theoretical result. Finally, compared with Chatterjee (2021) who showed that $$\widehat{\omega }_k$$ converges almost surely to $$\omega _k$$ as $$n \rightarrow \infty$$, our result (11) gives the nonasymptotic result.…”

“…Moreover, as emphasized in Section 3.1, ω𝑘 is able to measure the correlation for the continuous or binary response with the predictor 𝑋 (𝑘) , it implies that ( 10) is valid to retain important predictors for models (1) and ( 2). This is one of main differences from other existing methods (e.g., Fan & Lv, 2008;Li et al, 2012b;Xia & Li, 2021). In addition, unlike Ma et al (2017), the measure (9) in the XI-SIS method is a model-free approach and is able to handle ties in responses and predictors.…”

“…Li et al (2012b) proposed the model-free feature screening method based on the distance correlation (DC), which can be expressed in terms of the Pearson correlation coefficient. Xia & Li (2021) developed the copula-based partial correlation method. Chen (2021) and Chen and Yi (2022) extended the distance correlation to handle survival data with measurement error in predictors.…”