Minimax Robust Designs and Weights for Approximately Specified Regression Models with Heteroscedastic Errors
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异方差与回归模型近似设定下的极小极大稳健设计及权重
- 研究问题
- 当回归均值函数可能设定错误且误差可能异方差时，如何联合选择设计点与估计权重，控制最坏情形的拟合误差？
- 方法与思路
- 作者以两类偏离下的最大积分均方误差为目标，求解普通最小二乘、加权最小二乘及附加无偏约束的设计问题。
- 材料陈述的贡献
- 摘要称，给出极小极大设计及最优权重；在多项式示例中，无偏约束下的设计是经典 D 最优设计的平滑版本。
- 阅读关注点
- 核对偏离集合如何限定、积分误差的权重与设计空间，以及三类问题的适用范围、无偏约束和模拟比较的具体设定。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；不能推断后续刊本变化。前两类解限于所述无交互线性模型与球形空间，模拟收益不能泛化。
原文摘要
This article addresses the problem of constructing designs for regression models in the presence of both possible heteroscedasticity and an approximately and possibly incorrectly specified response function. Working with very general models for both types of departure from the classical assumptions, I exhibit minimax designs and correspondingly optimal weights. Simulation studies and a case study accompanying the theoretical results lead to the conclusions that the robust designs yield substantial gains over some common competitors, in the presence of realistic departures that are sufficiently mild so as to be generally undetectable by common test procedures. Specifically, I exhibit solutions to the following problems: P1, for ordinary least squares, determine a design to minimize the maximum value of the integrated mean squared error (IMSE) of the fitted values, with the maximum being evaluated over both types of departure; P2, for weighted least squares, determine both weights and a design to minimize the maximum IMSE, and P3, choose weights and design points to minimize the maximum IMSE, subject to a side condition of unbiasedness. Solutions to P1 and P2 are given for multiple linear regression with no interactions and a spherical design space. For P3 the solution is given in complete generality; as an example, I consider polynomial regression. In this case the minimax design turns out to be a smoothed version of the D-optimal design, with modes coinciding exactly with the support points of this classical design.