Abstract

This article develops new tools to quantify uncertainty in optimal decision making and to gain insight into which variables one should collect information about given the potential cost of measuring a large number of variables. We investigate simultaneous inference to determine if a group of variables is relevant for estimating an optimal decision rule in a high-dimensional semiparametric framework. The unknown link function permits flexible modeling of the interactions between the treatment and the covariates, but leads to nonconvex estimation in high dimension and imposes significant challenges for inference. We first establish that a local restricted strong convexity condition holds with high probability and that any feasible local sparse solution of the estimation problem can achieve the near-oracle estimation error bound. We further rigorously verify that a wild bootstrap procedure based on a debiased version of the local solution can provide asymptotically honest uniform inference for the effect of a group of variables on optimal decision making. The advantage of honest inference is that it does not require the initial estimator to achieve perfect model selection and does not require the zero and nonzero effects to be well-separated. We also propose an efficient algorithm for estimation. Our simulations suggest satisfactory performance. An example from a diabetes study illustrates the real application. Supplementary materials for this article are available online.

摘要

本文开发了新工具来量化最佳决策中的不确定性，并深入了解在给定测量大量变量的潜在成本的情况下应该收集哪些变量的信息。我们研究同时推理以确定一组变量是否与估计高维半参数框架中的最佳决策规则相关。未知链接函数允许对治疗和协变量之间的相互作用进行灵活建模，但会导致高维的非凸估计，并对推理提出重大挑战。我们首先确定局部受限的强凸性条件以高概率成立，并且估计问题的任何可行的局部稀疏解都可以达到近似预言的估计误差界限。我们进一步严格验证，基于局部解的去偏版本的 wild bootstrap 过程可以为一组变量对最优决策的影响提供渐近诚实的统一推理。诚实推理的优点是它不需要初始估计器来实现完美的模型选择，也不需要很好地分离零和非零效应。我们还提出了一种有效的估计算法。我们的模拟表明性能令人满意。糖尿病研究的一个例子说明了实际应用。本文的补充材料可在线获取。