Journal
ANNALS OF STATISTICS
Volume 41, Issue 1, Pages 342-369Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/13-AOS1087
Keywords
Feature screening; high dimension; polynomial splines; quantile regression; randomly censored data; sure independence screening
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Funding
- NSF [DMS-10-07396, DMS-10-07603]
- NIH [R01GM080503]
- National Natural Science Foundation of China [11129101]
- Baruch College Eugene M. Lang Fellowship
- Direct For Mathematical & Physical Scien [1237234] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [1237234] Funding Source: National Science Foundation
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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.
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