High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local heritability. Group inference in regression models can be measured with respect to a weighted quadratic functional of the regression sub-vector corresponding to the group. Asymptotically unbiased estimators of these weighted quadratic functionals are constructed and a novel procedure using these estimators for inference is proposed. We derive its asymptotic Gaussian distribution which enables the construction of asymptotically valid confidence intervals and tests which perform well in terms of length or power. The proposed test is computationally efficient even for a large group, statistically valid for any group size and achieving good power performance for testing large groups with many small regression coefficients. We apply the methodology to several interesting statistical problems and demonstrate its strength and usefulness on simulated and real data.
Z. Guo was supported in part by the NSF grants DMS-1811857, DMS-2015373 and NIH-1R01GM140463-01; Z. Guo also acknowledges financial support for visiting the Institute of Mathematical Research (FIM) at ETH Zurich. P. Bühlmann was supported in part by the European Research Council under the Grant Agreement No 786461 (CausalStats - ERC-2017-ADG). T.T. Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH grants R01-GM129781 and R01-GM123056.
Z. Guo is grateful to Dr. Cun-Hui Zhang, Dr. Hongzhe Li, and Dr. Dave Zhao for helpful discussions.
Zijian Guo. Claude Renaux. Peter Bühlmann. Tony Cai. "Group inference in high dimensions with applications to hierarchical testing." Electron. J. Statist. 15 (2) 6633 - 6676, 2021. https://doi.org/10.1214/21-EJS1955