Minimax Robust Designs and Weights for Approximately Specified Regression Models with Heteroscedastic Errors
研究了在响应函数近似指定且可能存在异方差时，如何构造极小极大设计及最优权重，通过模拟和案例表明该设计在常见偏离下优于传统方法。
Abstract 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.