观测值相关的线性模型的最优设计
Optimal design for linear models with correlated observations
摘要：
在一般的线性回归模型中，考虑了观测值相关时最小二乘估计的最优设计的确定问题。给出了给定设计最优性的一个必要条件，将误差不相关模型中最优设计的经典等价理论推广到相依数据的情况。如果回归函数是由协方差核定义的积分算子的本征函数，则相应的测度定义了一个普遍的最优设计。对于几个模型，普遍的最优设计可以明确地确定。特别地，证明了均匀分布对于一类具有广泛协方差核的三角回归模型是普遍最优的，而反正弦分布对于具有对数势定义的相关结构的多项式回归模型是普遍最优的.据作者所知，这些研究结果提供了第一个明确的结果最优设计的回归模型与相关的意见，这并不限于位置规模模型。
英文摘要：
In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. If the regression functions are eigenfunctions of an integral operator defined by the covariance kernel, it is shown that the corresponding measure defines a universally optimal design. For several models universally optimal designs can be identified explicitly. In particular, it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.
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