Research On Several Problems Of Optimal Experimental Designs Based On R-criterion

Scientific experiment is an important vehicle for people to recognize and understand nature.Experimental design is a key branch of statistics,which can substantially improve the accuracy and efficiency of the statistical analysis via the data that is achieved by optimally arranging the experimental schemes.Optimal design is one of the main research fields of experimental design,the origin of which can be traced back to,a century ago,the pioneering work by Smith(1918).Wald(1943)proposed the famous D-optimality criterion by using the accuracy of the parameters estimation as a standard for measuring the performance of design schemes.With the development of the theory of optimal designs,there are various optimality criteria that have been formulated by the analyst,depending on the different purposes of the experiment,such as A-optimality,E-optimality,R-optimality,T-optimality and IMSE-optimality,etc.To further perfect and enrich these results,we will in this paper investigate the problem of optimal designs for several common used models based on the R-optimality criterion,including Fourier regression models,random coefficients regression models,regression models with the asymmetric errors,and linear and nonlinear multi-factor models.In the first chapter,a brief introduction about the basic theories of optimal experi-mental designs for the ordinary linear models is given,and a historical overview of the R-optimality criterion is introduced.The second chapter studies the problem of constructing R-optimal designs for Fourier regression in linear models.When a set of estimable functions is of interest,we give the definition of the R-optimality criterion under a full rank subsystem and state the corresponding general equivalence theorem.In a partial cycle,we obtain the R-optimal designs for the first-order Fourier regression and further discuss the relative R-efficiency of the equidistant sampling.In a complete cycle,the R-optimal designs for estimating the specific pairs of the parameters in Fourier regression with larger order are derived explicitly.In the third chapter,we study optimal designs for the R-criterion in random coeffi-cients regression within linear mixed effects models.For the prediction of the individual parameters,the definition of the R-optimality criterion based on the mean squared error matrix of the predictor is given,and the general equivalence theorem is provided.For the prediction of the individual deviations,however,this direct definition makes sense in only the case of the random effects with positive definite covaiance matrix.Accordingly,an analytical result is also derived for the characterization of R-optimal designs for predic-tion.Finally,we present several examples that illustrate the theoretical results obtained in this chapter.The fourth chapter studies the problem of R-optimal designs for both linear and nonlinear regression models with asymmetric errors.We propose a new class of R-optimality criterion based on the second-order least squares estimator,and establish the general equivalence theorem for R-optimality.Moreover,several invariance properties of R-optimal designs are investigated.The numerical results show that the number of support points of R-optimal designs may be greater than the number of unknown parameters.In the fifth and sixth chapters,we concentrate on multi-factor models.The fifth chapter studies the R-optimal design problem for multi-factor linear regression models with heteroscedastic errors.For the multi-factor Kronecker product model,we show that a R-optimal design can be constructed by the product of the R-optimal designs for the marginal one-factor models.For the multi-factor additive models,R-optimal designs can also be achieved from the product designs for the one-factor models if the sufficient conditions are satisfied.Specifically,such designs are R-optimal only within the class of product designs when the additive models include a constant term but the orthogonality assumption on the marginal regression functions is unsatisfied.The sixth chapter studies the R-optimal design problem for a class of multiple regression models with information driven by the linear predictor.When the design region is a polyhedron,the bound of the number of support points of R-optimal designs and the optimal weights for the saturated R-optimal design are obtained firstly.Moreover,the construction method of R-optimal designs for such models is given.Finally,the obtained theoretical results have been verified by using two examples regarding the Poisson regression model and the proportional hazards model with censored data.