We propose a test of many zero parameter restrictions in a high dimensional linear iid regression model with $k$ $>>$ $n$ regressors. The test statistic is formed by estimating key parameters one at a time based on many low dimension regression models with nuisance terms. The parsimoniously parametrized models identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, it does not require a sparsity assumption nor therefore a regularized estimator, it is computationally fast compared to, e.g., de-biased Lasso, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error. We provide a parametric wild bootstrap for p-value computation, and prove the test is consistent and has non-trivial $\sqrt{n/\{\ln (n)\mathcal{M}% _{n}\}}$-local-to-null power where $\mathcal{M}_{n}$ is the $l_{\infty }$ covariate fourth moment.

翻译：我们提出了一种检验方法，用于高维线性独立同分布回归模型中多个参数为零的约束，其中回归变量个数 $k$ 远大于样本量 $n$。该检验统计量通过基于多个包含干扰项的低维回归模型逐一估计关键参数来构造。这些简约参数化模型能够识别原始感兴趣参数是否为零。估计固定的低维子参数可提高估计精度，无需稀疏性假设，因此也无需正则化估计量；与例如去偏Lasso相比，计算速度快；并且仅使用一系列加权估计量中的最大值，降低了检验统计量的复杂度，进而减少了估计误差。我们提供了参数化野自助法用于p值计算，并证明了该检验的一致性，以及其具有非平凡的 $\sqrt{n/\{\ln (n)\mathcal{M}_{n}\}}$-局部备择功效，其中 $\mathcal{M}_{n}$ 是协变量的 $l_{\infty}$ 四阶矩。