Inference for Large‐Scale Linear Systems With Known Coefficients
研究如何检验一个可能欠定的线性方程组是否存在非负解，提出一种仅需解线性规划即可实现的检验方法，适用于高维应用场景。
This paper considers the problem of testing whether there exists a non‐negative solution to a possibly under‐determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high‐dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size.