Minimax Efficient Random Experimental Design Strategies With Application to Model-Robust Design for Prediction

In game theory and statistical decision theory, a random (i.e., mixed) decision strategy often outperforms a deterministic strategy in minimax expected loss. As experimental design can be viewed as a game pitting the Statistician against Nature, the use of a random strategy to choose a design will often be beneficial. However, the topic of minimax-efficient random strategies for design selection is mostly unexplored, with consideration limited to Fisherian randomization of the allocation of a predetermined set of treatments to experimental units. Here, for the first time, novel and more flexible random design strategies are shown to have better properties than their deterministic counterparts in linear model estimation and prediction, including stronger bounds on both the expectation and survivor function of the loss distribution. Design strategies are considered for three important statistical problems: (i) parameter estimation in linear potential outcomes models, (ii) point prediction from a correct linear model, and (iii) global prediction from a linear model taking into account an <i>L</i>₂-class of possible model discrepancy functions. The new random design strategies proposed for (iii) give a finite bound on the expected loss, a dramatic improvement compared to existing deterministic exact designs for which the expected loss is unbounded. Supplementary materials for this article are available online.

在博弈论与统计决策论中，极小化极大期望损失（minimax expected loss）准则下，随机（即混合）决策策略往往优于确定性决策策略。由于可将实验设计视为统计学家与自然之间的对抗博弈，采用随机策略选取实验设计往往更具优势。然而，有关设计选取的极小化高效随机策略的研究大多尚未展开，现有探讨仅局限于将预先设定的处理集分配至实验单元的费希尔随机化（Fisherian randomization）方法。本文首次证明，相较于其确定性对应策略，新颖且更具灵活性的随机设计策略在线性模型估计与预测任务中表现更优，包括可为损失分布的期望与生存函数（survivor function）均提供更紧的界。本文针对三类重要统计问题展开了设计策略研究：(i) 线性潜在结果模型中的参数估计；(ii) 基于正确设定线性模型的点预测；(iii) 考虑L₂类潜在模型偏差函数的线性模型全局预测。针对问题(iii)提出的新型随机设计策略可将期望损失限定为有限值，相较于现有确定性精确设计（其期望损失无界），这是一项突破性改进。本文补充材料可在线获取。