UNIFORM-IN-SUBMODEL BOUNDS FOR LINEAR REGRESSION IN A MODEL-FREE FRAMEWORK
研究了高维数据中经模型选择后线性回归估计量的误差和线性表示界，这些界对子模型集合一致成立，且不依赖模型假设，适用于独立和相关数据，有助于解释变量缩减后的回归结果并证明模型选择后推断的合理性。
For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a “submodel”) where classical linear regression applies. We analyze linear regression estimators resulting from model selection by proving estimation error and linear representation bounds uniformly over sets of submodels. Based on deterministic inequalities, our results provide “good” rates when applied to both independent and dependent data. These results are useful in meaningfully interpreting the linear regression estimator obtained after exploring and reducing the variables and also in justifying post-model-selection inference. All results are derived under no model assumptions and are nonasymptotic in nature.