STATISTICAL INFERENCE WITH F-STATISTICS WHEN FITTING SIMPLE MODELS TO HIGH-DIMENSIONAL DATA
研究了高维数据下简单线性子模型的F检验，证明即使模型设定错误，标准F检验在渐近意义上仍然有效，无需对总体参数施加限制或假设数据正态。
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.