异方差随机系数回归模型的最优设计

最优设计方法在工程技术领域和工农业生产中具有广泛的应用．随机系数模型的最优设计研究中通常假定随机误差项具有相同的方差，实际中误差的产生往往与观测点有关，从而具有异方差性质．本文研究一般闭区间设计域上异方差随机系数回归模型的最优近似设计问题．我们获得了最优设计可以在设计域的两个端点处得到的一组充分条件，并进一步证明了当误差项方差具有对称结构且设计域是对称区间时，设计域两个对称端点处的等权重设计同时具有多重最优性质，这时最优设计不依赖于模型中随机误差项的方差结构及随机系数项的方差．

Optimal Designs of Heteroscedastic Random Coefficient Regression Models

The optimal design method has been widely used in the fields of engineering, and industrial and agricultural production. The study about optimal designs of random coefficient regression model is often done under the assumption that random errors are homoscedastic, but random errors are usually related to observation points and thus have heteroscedasticity. We consider the optimal approximate design of heteroscedastic random coefficient regression model on the closed intervals in this paper. Sufficient conditions are established to ensure that the optimal designs can be achieved at two extreme points of the design region. Equal-weight design on the extreme points of symmetrical design region is proved to be multiple optimal for the random coefficient regression model with symmetrical heteroscedastic errors. It doesn't depend on the heteroscedastic structure or variances of random coefficients in the model.