Multiple Response Linear Model Testing And Discrimination Of Experimental Design

In optimum design theory, many results have been obtained under the assumption of exactly correct response and homoscedasticity. But this assumption is not always correct. In most situations, we don't know the correctness of the chosen model and are always confronted with the choice of some models. It's not in reason for the tradition optimum design method in this situation. For this case, it's need to test the model before fitting. Certainly, the power of the test will be max under the optimal design. Otherwise, we are always confronted with the choice of some models so we should generate design to discriminate these models. In this paper, we, in this situation, extend the results from single-response linear model to multi-response linear models for the consummation of the optimum design theory.In Chapter 1, before the discussion for the experiment designs of the multi-response linear models, we introduce some result of the single-response linear model in this chapter. Based on these results, we will extend it from single-response linear model to multi-response linear models by the linear combination method. When don't know the correctness of the chosen model, we can generate designs for test, discrimination or forecast. Wiens( 1991), Jones and Mitchell(1978) have discussed the experiment designs for the test for lack of fit of single-response linear model. The optimality properties of uniform designs also have been proved by Wiens(1991). On the other hand, Atkinson(1975) generate the criterion, equivalence theorem and T-optimal design for the discrimination between models. As we all know, the discrimination between two models is the basic. Otherwise, the response forecast is often a keystone of experiment designs. Welth(1983) discussed the generation of the optimal design under the MSE criterion but it's pity that there were no result for the multi-response linear models.In Chapter 2, after the introduction of the Wiens(1991)'s result, the optimality properties of uniform designs, in chapter 1, it's nature that we guess the optimality properties of uniform designs also exist in multi-response linear models. In this chapter, we predigest the matrix calculation by model transform of Khuri(1985). What's more, we generate the definiens of lack of fit not only for the originally models but also for the transform models. At last, we prove the optimality properties of uniform designs for the multi-response linear models. In a short, when the model is lack of fit, the uniform designs maximin the minimum power of the test and when the model is not lack of fit, the uniform designs minimum the maximin power of the test. Certainly, the method of the test in this chapter is Khuri(1985) and the uniform designs is optimal under this kind of test. Contrarily, it's also show the efficiency of this test.In Chapter 3, experiment designs for the discrimination between two multi-response linear...