關於多項式迴歸模型下之最適設計問題己經在很多文獻中被討論到。對於定義在[a,b]之多項式迴歸型,其正合D-最適設計的最小樣本數Huang (1987)和Gaffke (1987)都給出了相似的充分條件。在本文中我們則是對一混合實驗模型作探討。一混合實驗為一包含q個非負成分${x_1,...,x_q}$,且 $sum_{i=1}^q x_i =1$的q-1維之機率空間$S^{q-1}$ 上的實驗設計。Kiefer (1961)證明了在Scheffeˊ的二次混合實驗模型下之D-最適設計,而基於此一結果我們證明2維與3維在Scheffe的二次混合實驗模型下之正合D-最適設計,並對於4維 至9維的模型給出一些數值的結果。

The exact D-optimal design problems for regression models has been in-vestigated in many literatures. Huang (1987) and Gaffke (1987) provided
a sufficient condition for the minimum sample size for an certain set of
candidate designs to be exact D-optimal for polynomial regression models
on a compact interval. In this work we consider a mixture experiment with
q nonnegative components, where the proportions of components are sub-
ject to the simplex restriction $sum_{i=1}^q x_i =1$, $x_i ≧ 0$. The exact D-optimal designs for mixture experiments for Scheffe’s quadratic models are investigated. Based on results in Kiefer (1961) results about the exact D-optimal designs for mixture models with two or three ingredients are provided and numerical verifications for models with ingredients between four and nine are presented.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Lists of Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Two and three ingredients . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Two ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Three ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

4 Four or more ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

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