Abstract

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 L₂-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.

摘要

在博弈论和统计决策理论中，随机（即混合）决策策略在 minimax 预期损失方面通常优于确定性策略。由于实验设计可以被视为统计学家与自然的博弈，使用随机策略来选择设计通常是有益的。然而，用于设计选择的最小最大效率随机策略的主题大多是未探索的，考虑仅限于将一组预定处理分配给实验单元的费希尔随机化。在这里，首次展示了新颖且更灵活的随机设计策略在线性模型估计和预测中比其确定性策略具有更好的属性，包括对损失分布的期望和幸存函数的更强限制。L ₂类可能的模型差异函数。为 (iii) 提出的新随机设计策略给出了预期损失的有限界限，与预期损失无界的现有确定性精确设计相比，这是一个显着的改进。本文的补充材料可在线获取。