标题： 一种用于检验一般高维假设的投影追踪框架 显示英文标题

标题： A projection pursuit framework for testing general high-dimensional hypothesis

摘要： 本文发展了一种框架用于检验高维模型中的广义假设，其中变量的数量可能远超观察值的数量。现有文献仅考虑了少数几种假设，例如检验模型参数的单个坐标。然而，检验广义和复杂假设的问题仍然广泛未解决。我们提出了一种新的推断方法，围绕假设自适应投影追踪框架展开，解决了最一般情况下的检验问题。所提出的推断以一类新定义的估计量为中心，该估计量被定义为未知量初始猜测在零假设所定义空间上的 $l_1$投影。此投影自动考虑了零假设的结构，并允许我们研究许多长期存在的问题的形式化推断。例如，我们可以直接对模型参数的稀疏性水平和最小信号强度进行推断。鉴于前者是高维统计理论发展的基础条件之一，而后者是建立变量选择性质的关键条件，这尤其具有重要意义。此外，所提出的方法在渐近意义上是精确的，并且对于检验高维参数的非常广泛的泛函具有令人满意的功效属性。模拟研究进一步支持了我们的理论主张，并展示了所提出检验在有限样本下的良好规模和功效属性。 显示英文摘要

摘要： This article develops a framework for testing general hypothesis in high-dimensional models where the number of variables may far exceed the number of observations. Existing literature has considered less than a handful of hypotheses, such as testing individual coordinates of the model parameter. However, the problem of testing general and complex hypotheses remains widely open. We propose a new inference method developed around the hypothesis adaptive projection pursuit framework, which solves the testing problems in the most general case. The proposed inference is centered around a new class of estimators defined as $l_1$ projection of the initial guess of the unknown onto the space defined by the null. This projection automatically takes into account the structure of the null hypothesis and allows us to study formal inference for a number of long-standing problems. For example, we can directly conduct inference on the sparsity level of the model parameters and the minimum signal strength. This is especially significant given the fact that the former is a fundamental condition underlying most of the theoretical development in high-dimensional statistics, while the latter is a key condition used to establish variable selection properties. Moreover, the proposed method is asymptotically exact and has satisfactory power properties for testing very general functionals of the high-dimensional parameters. The simulation studies lend further support to our theoretical claims and additionally show excellent finite-sample size and power properties of the proposed test.