在本論文中，我們探討在混合實驗上的取點問題。一混合實驗，是在一包含q個非負的成分{xi,i=1,...,q}的(q-1) 維之單純型機率空間S^{q-1}上所設計的實驗，並且以單純的限制式sum_{i=1}^q x_i=1為實驗的條件。在此，我們研究當混合實驗的模型屬於Scheffe（1958）所定義模型集合中，但並不確定其模型究竟屬於線性、二次或沒有交互作用之三次的情況下，如何尋找一穩健D型最適設計的問題。在不確知Scheffe模型的型態時，我們希望利用個別模型下之D型最適設計，來尋找出相關的穩健D型最適設計。若不確定實驗的模型是Scheffe的線性或二次之下，我們得證其穩健D型最適設計為個別D型最適設計的最佳凸組合。針對在考慮Scheffe線性和沒有交互作用之三次模型的情況，相關的穩健D型最適設計有一些數值上的驗證和推測結果，對線性、二次和沒有交互作用的三次模型，我們也有類似的結果。最後，我們討論在給定Scheffe線性和二次模型下之穩健D型最適設計，探討其在小中取大準則下的效率Dr問題。

A mixture experiment is an experiment in which the
q-ingredients {xi,i=1,...,q} are nonnegative and subject to the simplex restriction sum_{i=1}^q x_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. In this work, we investigate the robust D-optimal designs for mixture experiments with consideration on uncertainties in the Scheffe's linear, quadratic and cubic model without 3-way effects. The D-optimal designs for each of the Scheffe's models are used to find the robust D-optimal designs. With uncertianties on the Scheffe's linear and quadratic models, the optimal convex combination of the two model's D-optimal designs can be proved to be a robust D-optimal design. For the case of the Scheffe's linear and cubic model without 3-way effects, we have some numerical results about the robust D-optimal designs, as well as that for Scheffe's linear, quadratic and cubic model without 3-way effects. Ultimately, we discuss the efficiency of a maxmin type criterion D_r under given the robust D-optimal designs for the Scheffe's linear and quadratic models.

1 Introduction
2 Preliminary for the robust D-optimal design
2.1 Cmplete classes for second degree mixture models
2.2 Invariant symmetric block matrices
2.3 Generalized D-optimal equivalence theorem
3 Robust D-optimal designs for the Scheffe's linear and quadratic model
3.1 Robust D-optimal designs
3.2 Robust D-optimal designs for Scheffe's models
3.3 Dr-efficiency of robust D-optimal design
4 Some other results and discussion

Appendix
A Scheffe models
B Exampe of exchangeable designs
C Plot of Dr-efficiency for given r'=0.67 at different q
D Plot of Dr-efficiency under different r for q=2,...,250

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