一個混合實驗，是在一包含q個非負的成分{x_i, i=12,…,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ㄧ單純的限制式Σx_i=1為實驗的條件。雖然在許多混合實驗的應用上其觀測值具有相關性存在，但通常觀測值會被假設為無相關性。在本論文中，我們研究在有相關性觀測值下，其ㄧ般的最小平方法估計量和高斯馬可夫法估計量之間的差異性；並且得知在某些混合模型與ㄧ特殊的共變異數結構下，對於ㄧ般的最小平方法估計量和高斯馬可夫法估計量兩者的未知參數向量是相同的；此外，從無相關性觀測值的D型和A型最適設計，可能會得到對應於有相關性觀測值的最適設計。在此，我們所討論的模型包含Scheff'e模型、對數對比模型、包含同質函數的模型以及包含反向項的模型。

A mixture experiment is an experiment in which the q-ingredients {x_i,i=1,2,...,q} are nonnegative and ubject to the simplex restriction Σx_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. It is usually assumed that the observations are uncorrelated, although in many applications the observations are correlated. We study the difference between the
ordinary least square estimator and the Gauss Markov estimator under correlated observations. It is shown that for certain models and a special covariance structure for the mixture experiments, the
unknown parameter vector for the ordinary least square estimators and the Gauss Markov estimators are the same. Moreover, we also show that the corresponding optimal designs may be obtained from previous D- and A-optimal designs for uncorrelated observations. The models studied here includ Scheff'e models, log contrast models, models containing homogeneous functions, and models containing inverse terms.

Contents
Abstract i

1 Introduction 1

2 Ordinary least square estimators and generalized least square estimators under polynomial models for mixture experiments 3

3 Ordinary least square estimators and generalized least square estimators under log contrast mixture models 8
3.1 Estimation under linear log contrast model . . . . . . . . . . . . . . . . . . . . . 8
3.2 Estimation under quadratic log contrast model . . . . . . . . . . . . . . . . . . . 10
3.3 Estimation under linear log contrast model without intercept . . . . . . . . . . . 14

4 Conclusions and Discussions 18

References 19

Appendix 20
A.1 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
A.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A.3 Proof of Theorem 3.3 and Theorem 3.8 . . . . . . . . . . . . . . . . . . . . . . 22
A.4 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A.5 Some useful theorems and optimal designs for linear log contrast model with uncorrelated observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.6 Support points designs and corresponding information matrices with maximum D-values for correlated observations under model (3.20) for the sample size n =k + 1 with restricted value
a = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.7 Box plots for different number of interested ingredients and covariances under model (3.20) for the sample size n = k + 1 with restricted value
a = 3 . . . . . . 33

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Accepted by Journal of Statistical Planning and Inference.