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High-dimensional group inference is an essential part of statistical methodsfor analysing complex data sets, including hierarchical testing, tests ofinteraction, detection of heterogeneous treatment effects and inference forlocal heritability. Group inference in regression models can be measured withrespect to a weighted quadratic functional of the regression sub-vectorcorresponding to the group. Asymptotically unbiased estimators of theseweighted quadratic functionals are constructed and a novel procedure usingthese estimators for inference is proposed. We derive its asymptotic Gaussiandistribution which enables the construction of asymptotically valid confidenceintervals and tests which perform well in terms of length or power. Theproposed test is computationally efficient even for a large group,statistically valid for any group size and achieving good power performance fortesting large groups with many small regression coefficients. We apply themethodology to several interesting statistical problems and demonstrate itsstrength and usefulness on simulated and real data.

Group Inference in High Dimensions with Applications to Hierarchical Testing