We introduce a quantile-adaptive framework for nonlinear variable screening with high-dimensional heterogeneous data. This framework has two distinctive features: (1) it allows the set of active variables to vary across quantiles, thus making it more flexible to accommodate heterogeneity; (2) it is model-free and avoids the difficult task of specifying the form of a statistical model in a high dimensional space. Our nonlinear independence screening procedure employs spline approximations to model the marginal effects at a quantile level of interest. Under appropriate conditions on the quantile functions without requiring the existence of any moments, the new procedure is shown to enjoy the sure screening property in ultra-high dimensions. Furthermore, the quantile-adaptive framework can naturally handle censored data arising in survival analysis. We prove that the sure screening property remains valid when the response variable is subject to random right censoring. Numerical studies confirm the fine performance of the proposed method for various semiparametric models and its effectiveness to extract quantile-specific information from heteroscedastic data.

Quantile-adaptive model-free variable screening for high-dimensional heterogeneous data

This article considers ultrahigh dimensional prediction problems with censored response vari- ables. We propose a two-step model averaging procedure for improving prediction accuracy of the true conditional mean of a censored response variable. The first step is to construct a class of candidate models, each with low-dimensional covariates. For this, a feature screening procedure is developed to separate the active and inactive predictors through a fused mean- variance index and group covariates with similar size of index together to form regression models with censored response variables. The new model-free screening method can easi- ly deal with many types of predictors and response variables, such as discrete, categorical and continuous variables, still works well when predictors have heavy-tailed distributions or strongly dependend on each other, and enjoys rank consistency properties under mild regularity conditions. The second step is to find the optimal model weights for averaging by adapting a delete-one Mallows criterion, where the standard constraint that weights sum to one is removed. The theoretical results show that the delete-one Mallows criterion achieves the lowest possible prediction loss asymptotically. Numerical studies demonstrate the su- perior performance of the proposed variable screening and model averaging procedures over existing methods.

Zhao Xing-Qiu

Papers

Optimal Model Averaging Estimation in High-dimensional Censored Linear Models