Optimal minimax random designs for weighted least squares estimators
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加权最小二乘估计的最优极小极大随机设计
- 研究问题
- 真实回归函数未必线性时，如何选择随机预测变量取值，使加权最小二乘有效估计其最佳线性近似并控制最坏风险？
- 方法与思路
- 作者以最佳线性近似为目标构造加权最小二乘的渐近极小极大准则，求解随机设计，并与既有确定性设计比较。
- 材料陈述的贡献
- 摘要称，该框架下普通最小二乘通常不一致，固定设计的极小极大风险无限；最优随机设计在噪声方差增大时趋于确定性设计。
- 阅读关注点
- 应核对最佳线性近似的参照分布、权重构造、函数类与风险定义，以及固定设计风险无限和两类设计重合的条件。
- 阅读边界
- 仅依据所给 Crossref 摘要；不一致性和无限风险均依赖文中框架，不能普遍化。所述一般化位于补充材料，摘要未给细节。
原文摘要
Summary This work studies an experimental design problem where the values of a predictor variable, denoted by $x$, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$, but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$ values should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that, when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and a generalization is given in the Supplementary Material.