Collaborative Research: Higher-Order Asymptotics and Accurate Inference for Post-Selection

许多统计分析使用模型选择程序。也许最常见的模型选择问题是线性回归中的变量选择问题。选择的原则性动机包括对可解释性的渴望，防止过度匹配，以及对统计权力的担忧。当数据是高维的，具有更多的解释变量而不是观察时，就会产生实际的动机。相关应用横跨科学研究的整个领域，从神经科学、医学和物理学，到经济学、社会学和心理学。现已有一大批变量选择程序可用，统计界已将重点转向选择后的推断问题。当使用相同的数据来选择一个模型并对该模型进行推断时，统计推断的标准方法不再有效。拥有准确的选择后推理程序至关重要，这些程序也足够强大，足以检测观察到的偏离科学假设的情况，并避免在有限样本中进行准确推理所需的强分布假设。这项研究旨在开发既准确又强大的选择后推理方法，特别强调减少依赖于样本量的统计误差。本项目的目标是进一步了解后选择推理的渐近理论，特别是基于CovTest和截断高斯(Tg)统计量的选择性推理，以及使用后选择区间(POSI)过程的同时推理。前两个过程可以由选择性误差控制来激励，即，对所选模型参数的误差控制。POSI方法寻求控制所有可能的子模型参数的家族式错误率。虽然这些程序产生了有效的选择后推论，但如果没有强有力的假设，它们特别容易受到在小到中等样本量的现实环境中违反关键假设的影响，例如过于保守或不准确的可信区间，以及低功率。在这个项目中，将使用渐近展开、鞍点近似、Bootstrap以及高阶渐近中的相关技术来提高这些选择后推理过程的精度和能力。

Many statistical analyses utilize a model selection procedure. Perhaps the most common model selection problem is that of variable selection in linear regression. Principled motivations for selection include the desire for interpretability, prevention of over-fitting, and concerns about statistical power. A practical motivation arises when the data are high-dimensional, with more explanatory variables than observations. Relevant applications span the entire domain of scientific inquiry, from neuroscience, medicine and physics, to economics, sociology, and psychology. A large catalogue of variable selection procedures is now available, and the statistics community has turned its focus to the question of inference after selection. Standard methods of statistical inference are no longer valid when the same data are used to both select a model and make inferences about that model. It is fundamentally important to have accurate post-selection inference procedures that are also powerful enough to detect observed departures from scientific hypotheses and that avoid the strong distributional assumptions needed for exact inference in finite samples. This research aims to develop post-selection inference methodology that is both accurate and powerful, with particular emphasis on reducing statistical errors that depend on the sample size. The goal of this project is to further understanding of the asymptotic theory of post-selection inference, particularly selective inference based on the CovTest and truncated Gaussian (TG) statistic, as well as simultaneous inference using the post-selection intervals (PoSI) procedure. The first two procedures can be motivated by selective error control, i.e., error control for the selected model parameters. The PoSI method seeks to control family-wise error rates for all possible sub-model parameters. While these procedures yield valid post-selection inference, without strong assumptions they are particularly vulnerable to the effects of violation of key assumptions in the realistic setting of small to moderate sample sizes, such as overly-conservative or inaccurate confidence intervals, and low power. In this project, asymptotic expansions, saddle-point approximations, the bootstrap, and related techniques from higher-order asymptotics will be employed to improve accuracy and power for these post-selection inference procedures.