A simple yet powerful test for assessing goodness-of-fit of high-dimensional linear models

We evaluate the validity of a projection-based test checking linear models when the number of covariates tends to infinity, and analyze two gene expression datasets. We show that the test is still consistent and derive the asymptotic distributions under the null and alternative hypotheses. The asymptotic properties are almost the same as those when the number of covariates is fixed as long as p/n -> 0 with additional mild assumptions. The test dramatically gains dimension reduction, and its numerical performance is remarkable.

Automated summary

We evaluate the validity of a projection-based test for linear models when the number of covariates tends to infinity, showing that the test remains consistent and derives asymptotic distributions under the null and alternative hypotheses. The test gains dimension reduction significantly and demonstrates remarkable numerical performance, with asymptotic properties similar to when the number of covariates is fixed as long as p/n -> 0.