论文标题
多因素实验中的设计可采性和de la garza现象
Design admissibility and de la Garza phenomenon in multi-factor experiments
论文作者
论文摘要
确定给定回归问题的最佳设计是一个复杂的优化问题，尤其是对于具有多元预测变量的模型。设计可采性和不变性是减少优化问题复杂性的主要工具，并已成功应用于具有单变量预测指标的模型。特别是，几位作者为单变量模型中的饱和设计建立了足够的条件，其中最佳设计的支持点等于参数的数量。这些结果概括了著名的de la garza现象（1954年de la garza），该现象指出，对于程度$ p-1 $的多项式回归模型，任何最佳设计都可以基于$ P $点。本文首次为具有多元预测变量的模型提供了这些结果的扩展。特别是，我们研究了最佳设计的支持点的几何表征，以在具有多元预测变量的模型中为De la Garza现象的发生提供足够的条件，并以条件单变量回归模型中设计的可接受性来表征可允许设计的特性。
The determination of an optimal design for a given regression problem is an intricate optimization problem, especially for models with multivariate predictors. Design admissibility and invariance are main tools to reduce the complexity of the optimization problem and have been successfully applied for models with univariate predictors. In particular several authors have developed sufficient conditions for the existence of saturated designs in univariate models, where the number of support points of the optimal design equals the number of parameters. These results generalize the celebrated de la Garza phenomenon (de la Garza, 1954) which states that for a polynomial regression model of degree $p-1$ any optimal design can be based on at most $p$ points. This paper provides - for the first time - extensions of these results for models with a multivariate predictor. In particular we study a geometric characterization of the support points of an optimal design to provide sufficient conditions for the occurrence of the de la Garza phenomenon in models with multivariate predictors and characterize properties of admissible designs in terms of admissibility of designs in conditional univariate regression models.
