A- And R-optimal Designs For Hierarchical Linear Models With Heteroscedastic Errors

Hierarchical linear models have been widely concerned in the statistical field and practical application,but the designs of hierarchical linear model with heteroscedastic errors need more attention and further study at present.This paper mainly discusses the A-and R-optimal approximate designs of hierarchical linear model with heteroscedastic errors and aims at the accurate prediction of individual parameters.The work of optimal experimental designs are mainly to put forward optimal criterions and construct optimal designs for the model.The first part of this paper is to define A-and R-optimal criterions for hierarchical linear models with heteroscedastic errors for the accurate prediction of individual parameters.On the basis of proving the convexity of the criterion function,the equivalence theorem is established by means of directional derivative to describe A-and R-optimal designs,which provides a good theoretical tool for verifying the optimality of designs.Further,the efficiency of designs are introduced to compare designs,and the lower bound of the efficiency are discussed.In the second part,different heteroscedastic errors are selected to solve the analytical or numerical solutions of A-and R-optimal designs by taking random intercept regression model,random slope regression model and non-intercept quadratic model as examples.Then we compare the optimal design and the equireplicated design.The results show that equireplicated design for random intercept regression model is close to the optimal design,and less efficient than the optimal design for random slope regression model.It is convenient for us to choose the appropriate design before the experiment.In addition,this paper draws the conclusion that the A-optimal designs of the hierarchical linear models with heteroscedastic errors and homoscedastic errors are the same under special errors.For the quadratic model without random intercept,the range of optimal design points is further reduced.