Nonparametric screening for additive quantile regression in ultra-high dimension

Abstract

In practical applications, one often does not know the ‘true’ structure of the underlying conditional quantile function, especially in the ultra-high dimensional setting. To deal with ultra-high dimensionality, quantile-adaptive marginal nonparametric screening methods have been recently developed. However, these approaches may miss important covariates that are marginally independent of the response, or may select unimportant covariates due to their high correlations with important covariates. To mitigate such shortcomings, we develop a conditional nonparametric quantile screening procedure (complemented by subsequent selection) for nonparametric additive quantile regression models. Under some mild conditions, we show that the proposed screening method can identify all relevant covariates in a small number of steps with probability approaching one. The subsequent narrowed best subset (via a modified Bayesian information criterion) also contains all the relevant covariates with overwhelming probability. The advantages of our proposed procedure are demonstrated through simulation studies and a real data example.

Keywords:

• Additive model
• feature screening
• high dimension
• nonparametric quantile regression
• sure screening property
• variable selection

Mathematical Subject classifications:

• 62G08
• 62G99
• 62J02
• 62G05

Acknowledgments

The authors sincerely thank the Editor, Associate Editor, and anonymous referees for their valuable and constructive comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Li was supported by the 2022–2023 RSCA Award Program at California State University, Fullerton, United States.