A GEOMETRIC CHARACTERIZATION OF c-OPTIMAL DESIGNS FOR HETEROSCEDASTIC REGRESSION
A GEOMETRIC CHARACTERIZATION OF c-OPTIMAL DESIGNS FOR HETEROSCEDASTIC REGRESSION
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DOI：
10.1214/09-aos708
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发表时间：
2009-12-01
影响因子：
4.5
通讯作者：
Holland-Letz, Tim
中科院分区：
文献类型：
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作者：
Dette, Holger;Holland-Letz, Tim
我们考虑常见的非线性回归模型，其中方差和均值都是解释变量的参数函数。研究了均值和方差函数的参数都有意义时的c-最优设计问题。本文给出了c-最优设计的一个几何特征，推广了Elfving [Ann. Math. Statist. 23（1952）255-262]的最优设计。正如Elfving的著名刻画，c-最优设计可以被描述为凸集的边界点的表示。然而，在方差中出现感兴趣的参数的情况下，Elfving集的结构是不同的。粗略地说，对应于异方差回归模型的Elfving集是由基础模型诱导并由设计空间索引的椭球集合的凸船体。c-最优设计的特征在于向量c方向的直线与新Elfving集的边界相交的点的表示。该理论说明了几个例子，包括随机效应的药代动力学模型。
We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255-262] for c-optimal designs. As in Elfving's famous characterization, c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking, the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.