Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensional models is extremely challenging -- especially without making restrictive or unverifiable assumptions on the number of non-zero elements, i.e., the model sparsity. In this article, we propose a new methodology that transforms the original hypothesis into a moment condition and demonstrate that valid tests can be created without making any assumptions on the model sparsity. We formulate a restructured regression problem with the new features synthesized according to the null hypothesis directly; further, with the help of such new features, we have designed a valid test for the transformed moment condition. This construction enables us to test the null hypothesis, even if the original model cannot be estimated well. Although the linear tests in high dimensions are by nature very difficult to analyze we establish theoretical guarantees for Type I error control, allowing both the model and the vector representing the hypothesis to be non-sparse. The assumptions that are necessary to establish Type I error guarantees are shown to be weaker than the weakest known assumptions that are necessary to construct confidence intervals in the high-dimensional regression. Our methods are also shown to achieve certain optimality in detecting deviations from the null hypothesis. We demonstrate favorable finite-sample performance of the proposed methods, via both a numerical and a real data example.

本研究提出了一种在高维线性模型中测试线性假设的方法，可以不对模型的大小（即模型的稀疏性或表示假设的加载向量）进行任何限制，并通过测试与新设计的重组回归模型相关的时刻条件来实现。

稠密高维线性模型中的线性假设检验