Conditional quantile correlation learning for ultrahigh dimensional varying coefficient models and its application in survival analysis

Abstract

In this paper, we consider a robust approach to the ultrahigh dimensional variable screening under varying coefficient models. Different from the existing works focusing on the mean regression function, we propose a novel procedure based on the conditional quantile correlation sure independent screening (CQCSIS). This new proposal is applicable to heterogeneous or heavy-tailed data in general and is invariant to monotone transformation of the response. Furthermore, we generalize such a screening procedure to address censored lifetime data through inverse probability weighting. The CQCSIS can be easily implemented due to an application of nonparametric B-spline approxi-1 Statistica Sinica: Newly accepted Paper (accepted version subject to English editing) mation, and computed much faster than the kernel based screening method. Under some regularity conditions, we establish sure screening properties including screening consistency and ranking consistency for proposed approaches. In this paper we also attempt to construct a two-stage variable selection procedure for a further improvement of performance of CQCSIS based on a group SCAD penalization. Extensive simulation examples and real data applications are presented for illustration.

Nonparametric estimation of conditional quantile functions in the presence of irrelevant covariates

“…Variable or feature selection in highdimensional linear quantile regression has been extensively studied in the literature and various shrinkage and screening techniques have been introduced to identify these significant covariates(e.g., Wang, Wu and Li, 2012;Fan, Fan and Barut, 2014;Zheng, Peng and He, 2015;Ma, Li and Tsai, 2017). For extensions to high-dimensional nonparametric quantile regression, we refer to Fernández-Val (2011), He, Wang andHong (2013) and Xia, Li and Fu (2018). In this section, we consider a general nonparametric quantile regression setting which contains high-dimensional mixed continuous and discrete covariates.…”

“…The above theorem complements some existing sure screening properties in highdimensional quantile estimation (c.f., He, Wang and Hong, 2013;Ma, Li and Tsai, 2017;Xia, Li and Fu, 2018). An alternative kernel screening procedure is to conduct the leaveone-out kernel estimation for each marginal quantile regression and then use the datadriven CV method to determine the optimal smoothing parameter.…”

“…Furthermore, we generalize the model setting and methodology to the case when the dimension of covariates is large (growing with the sample size n) and introduce a two-step procedure: (i) use a kernel-based quantile screening technique to remove the irrelevant continuous and discrete covariates, and (ii) apply the CV criterion to those that survive in the first step of screening and further select the significant covariates and determine the optimal smoothing parameters. Note that the existing literature on variable or feature selection in highdimensional quantile regression only considers the case of purely continuous covariates (e.g., He, Wang and Hong, 2013;Ma, Li and Tsai, 2017;Xia, Li and Fu, 2018). The present paper considers a more general setting which contains both the discrete and continuous covariates.…”

Sparse boosting for high‐dimensional survival data with varying coefficients

“…Assuming that error term follows the normal distribution, the Wald P value for testing null hypothesis β j =0 can be obtained by survreg function in R. After sorting 20 000 P values in an ascending order, the top markers in the ranked list reflect the strongest covariate effects on the mean survival time. Following the recommendation by Fan and Lv and other references, we keep the top variables in the ranked list and screen out the rest of the covariates. Thus, the first 66 variables in the ranked list are selected to conduct downstream analysis.…”

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

“…To the best of our knowledge, there are very few works applying this classical dependence concept in high-dimensional setting. Xia et al (2018) proposed a robust conditional feature screening approach, however, their method performs only robustly against the response but not against the covariates. Another relevant recent work is Ma et al (2017).…”