Nonparametric screening under conditional strictly convex loss for ultrahigh dimensional sparse data

Abstract

Sure screening technique has been considered as a powerful tool to handle the ultrahigh dimensional variable selection problems, where the dimensionality $p$ and the sample size $n$ can satisfy the NP dimensionality $\log p=O(n^{a})$ for some $a>0$ [J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 (2008) 849–911]. The current paper aims to simultaneously tackle the “universality” and “effectiveness” of sure screening procedures. For the “universality,” we develop a general and unified framework for nonparametric screening methods from a loss function perspective. Consider a loss function to measure the divergence of the response variable and the underlying nonparametric function of covariates. We newly propose a class of loss functions called conditional strictly convex loss, which contains, but is not limited to, negative log likelihood loss from one-parameter exponential families, exponential loss for binary classification and quantile regression loss. The sure screening property and model selection size control will be established within this class of loss functions. For the “effectiveness,” we focus on a goodness-of-fit nonparametric screening (Goffins) method under conditional strictly convex loss. Interestingly, we can achieve a better convergence probability of containing the true model compared with related literature. The superior performance of our proposed method has been further demonstrated by extensive simulation studies and some real scientific data example.