Theory And Application Research On Optimal Experimental Designs Of Mixture Model With Heteroscedastic Errors

Mixture experimental designs have been widely used in industrial and agricul-tural production, scientific experiment and daily life. The researchers establish an appropriate mixture model according to the actual situation, formulate reasonable ex-perimental plan under the guidance of optimal design, obtain rich data scientifically and effectively, and fit the variation between experimental index and the components which we care about. The traditional mixture experimental designs are based on the assumption that the experimental errors are homoscedastic, but heteroscedastic errors are more accordant with practical circumstances. In this paper, we mainly study the optimal experimental designs of different mixture models under the assumption of heteroscedastic errors, and the application of mixture experimental designs in portfo-lio investment. These research have important theoretical significance and practical application value. The main contents of this thesis are as follows:When the experimental design area is the direct sum of two sub-design areas, we firstly prove that D-、A-、Iλ-、G- or linear optimal designs for general product models with heteroscedastic errors can be constructed from the corresponding optimal designs for sub-models, and extend the conclusion to the case of multi-sub models. Then, it is proved that the direct product of D-, A- or linear optimal designs for the two mixture models with heteroscedastic errors is the corresponding optimal designs for the product mixture models under given conditions when the experiment design areas are regular simplexes. Through the research of the direct product designs for product models, we can use the optimal design of each sub-model for constructing without directly finding the optimal designs to deal with the complex models.When the complex mixture system can be divided into two independent sub-systems, we study D-, A-and linear optimality of the direct sum designs for addi-tive mixture models with heteroscedastic errors. Firstly, the sufficient conditions are given so that D-, A- or linear optimal designs for additive mixture models with het-eroscedastic errors can be constructed from the direct sum of D-, A-or linear optimal designs for homogeneous sub-models. It is worth noting that the weights in direct sum design are different when we construct different optimal designs. Then, Another sufficient conditions are given so that D-, A-or linear optimal designs for additive mixture models with heteroscedastic errors can be constructed from the correspond-ing optimal designs for mixture sub-models, as soon as the additive mixture models are polynomials. Through the research of the direct sum designs for additive mixture models, we can also construct the optimal design for complex models conveniently and rapidly.When the experimental design area is a regular simplex which adds a single point at zero, we research the optimal designs for a special mixture model with het-eroscedastic errors. Under given conditions, it is proved that D- or A-optimal designs for the special mixture models with additional constraint conditions can be construct-ed from the corresponding optimal designs for the mixture models defined on regular simplexes.The method of mixture experimental design is mostly applied in natural sci-ence, and it is seldom used in economics. In this paper, we use the conclusions of optimal designs for additive mixture models with heteroscedastic errors in portfolio investment, and give the reasonable explanations of portfolio risk and portfolio re-turn from the optimality of mixture experimental designs and the heteroscedasticity of experimental errors. Firstly, we select a reasonable heteroscedastic mixture model according to the expected rate of return. Then, as investors’risk aversion, we study the experimental designs of minimizing the maximum risk from expected rate of return on portfolio investment. Finally, we calculate the investment proportional coefficient with minimum portfolio risk through some examples.