Inference for Linear Systems with Unknown Coefficients
未知系数线性系统的统计推断AI 解读 ↓
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未知系数线性系统的统计推断
- 研究问题
- 当线性方程组包括斜率在内的全部系数均未知时，如何检验是否存在满足非负约束的解，并服务于部分识别参数的置信集构造？
- 方法与思路
- 作者先刻画原假设在全变差距离下的闭包，以分析无法获得有效检验功效的情形，再据此构造基于样本分割的检验程序。
- 材料陈述的贡献
- 摘要称，所提检验在允许维度随样本量快速增长的条件下有效，且计算临界值无需模拟；作者另用模拟展示理论结果的实用性。
- 阅读关注点
- 应核对原假设闭包的具体刻画、不可检验情形、样本分割方式，以及检验有效性所需条件和维度增长限制如何对应应用模型。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核验。摘要未说明具体条件、维度增长率及功效保证，不能据此判断适用范围。
原文摘要
This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system of equations, including the slope coefficients, to be unknown. For this reason, we describe the linear system as having unknown (as opposed to known) coefficients. This hypothesis testing problem arises naturally when constructing confidence sets for possibly partially identified parameters in the analysis of nonparametric instrumental variables models, treatment effect models, and random coefficient models, among other settings. To rule out certain instances in which the testing problem is impossible, in the sense that the power of any test will be bounded by its size, we begin our analysis by characterizing the closure of the null hypothesis with respect to the total variation distance. We then use this characterization to develop novel testing procedures based on sample-splitting. We establish the validity of our testing procedures under weak and interpretable conditions on the linear system. An important feature of these conditions is that they permit the dimensionality of the problem to grow rapidly with the sample size. A further attractive property of our tests is that they do not require simulation to compute suitable critical values. We illustrate the practical relevance of our theoretical results in a simulation study.