We study linear subset regression in the context of the high-dimensional overall model $y = \vartheta +\theta ' z + \epsilon $ with univariate response y and a d-vector of random regressors z, independent of $\epsilon $. Here, “high-dimensional” means that the number d of available explanatory variables is much larger than the number n of observations. We consider simple linear submodels where y is regressed on a set of p regressors given by $x = M'z$, for some $d \times p$ matrix M of full rank $p < n$. The corresponding simple model, that is, $y=\alpha +\beta ' x + e$, is usually justified by imposing appropriate restrictions on the unknown parameter $\theta $ in the overall model; otherwise, this simple model can be grossly misspecified in the sense that relevant variables may have been omitted. In this paper, we establish asymptotic validity of the standard F-test on the surrogate parameter $\beta $, in an appropriate sense, even when the simple model is misspecified, that is, without any restrictions on $\theta $ whatsoever and without assuming Gaussian data.

我们在高维总体模型$y = \vartheta +\theta ' z + \epsilon $的背景下研究线性子集回归，其中单变量响应y和随机回归量z的d向量，独立于$\epsilon $。这里，“高维”是指可用解释变量的数量d远大于观测值的数量n。我们考虑简单的线性子模型，其中y在由$x = M'z$给出的一组p回归量上进行回归，对于满秩$p < n$的某些$d \times p$矩阵M。相应的简单模型，即$y=\alpha +\beta ' x + e$，通常通过对整体模型中的未知参数$\theta $施加适当的限制来证明其合理性；否则，这个简单的模型可能会被严重错误指定，因为相关变量可能已被省略。在本文中，我们在适当的意义上建立了对代理参数$\beta $的标准F检验的渐近有效性，即使简单模型被错误指定，即对$\theta $没有任何限制，并且没有假设高斯数据。