高维数据拟合简单模型时F统计量的统计推断
Statistical inference with F-statistics when fitting simple models to high-dimensional data
摘要：
我们研究线性子集回归的背景下，高维整体模型$y = \vartheta+θ' z + ε$与单变量响应$y$和$d$ -向量的随机回归$z$，独立于$ε$。这里，“高维”意味着可用解释变量的数量$d$远大于观测值的数量PDMATHAMR。我们考虑简单的线性子模型，其中$y$是在$x = M'z$给出的一组$p$回归量上回归的，对于一些满秩$p < n$的$d \times p$矩阵$M$。相应的简单模型，即，$y=α+β' x + e$，可以通过对整个模型中的未知参数$θ$施加适当的限制来证明;否则，这个简单的模型可能被严重错误地指定。在本文中，我们建立了渐近有效性的标准$F$ -测试的替代参数$β$，在适当的意义上，即使当简单的模型是错误的。
英文摘要：
We study linear subset regression in the context of the high-dimensional overall model $y = \vartheta+θ' z + ε$ with univariate response $y$ and a $d$-vector of random regressors $z$, independent of $ε$. 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 sub-models 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, i.e., $y=α+β' x + e$, can be justified by imposing appropriate restrictions on the unknown parameter $θ$ in the overall model; otherwise, this simple model can be grossly misspecified. In this paper, we establish asymptotic validity of the standard $F$-test on the surrogate parameter $β$, in an appropriate sense, even when the simple model is misspecified.
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