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URN etd-0205109-154131 Author Miao-kuan Huang Author's Email Address huangmk@math.nsysu.edu.tw Statistics This thesis had been viewed 5210 times. Download 1306 times. Department Applied Mathematics Year 2008 Semester 1 Degree Ph.D. Type of Document Language English Title Optimal Designs for Log Contrast Models in Experiments with Mixtures Date of Defense 2009-01-10 Page Count 96 Keyword Equivalence theorem Geometric-arithmetic means inequality Invariant symmetric block matrices Kiefer ordering Lagrange interpolation polynomial Loewner ordering Abstract A mixture experiment is an

experiment in which the k ingredients are nonnegative and subject

to the simplex restriction Σx_i=1 on the

(k-1)-dimensional probability simplex S^{k-1}. This dissertation

discusses optimal designs for linear and

quadratic log contrast models for experiments with

mixtures suggested by Aitchison and Bacon-Shone (1984),

where the experimental domain is restricted further as in Chan (1992).

In this study, firstly, an essentially complete

class of designs under the Kiefer ordering for linear log contrast

models with mixture experiments is presented. Based on the

completeness result, Φ_p-optimal designs for all p, -∞<p≤1 including D- and A-optimal are obtained, where

the eigenvalues of the design moment matrix are used. By using the

approach presented here, we gain insight on how these

Φ_p-optimal designs behave.

Following that, the exact N-point D-optimal designs for

linear log contrast models with three and four ingredients are

further investigated.

The results show that for k=3 and N=3p+q ,1 ≤q≤2, there is an exact

N-point D-optimal design supported at the points of S_1 (S_2)

with equal weight n/N, 0≤n≤p , and puts the remaining

weight (N-3n)/N uniformly on the points of S_2 (S_1) as evenly as

possible, where S_1 and S_2 are sets of the supports of the

approximate D-optimal designs. When k=4 and N=6p+q , 1 ≤q≤5, an exact N-point design which distributes the weights as

evenly as possible among the supports of the approximate D-optimal

design is proved to be exact D-optimal.

Thirdly, the approximate D_s-optimal designs for

discriminating between linear and

quadratic log contrast models for experiments with

mixtures are derived.

It is shown that for a symmetric subspace of the finite

dimensional simplex, there is a D_s-optimal design with the nice structure that

puts a weight 1/(2^{k-1}) on the centroid of this subspace and the remaining weight is

uniformly distributed on the vertices of the experimental domain.

Moreover, the D_s-efficiency of the D-optimal design for

quadratic model and the design given by Aitchison and Bacon-Shone

(1984) are also discussed

Finally, we show that an essentially complete class of designs under

the Kiefer ordering for the quadratic log contrast model is the set

of all designs in the boundary of T or origin of T

. Based on the completeness result, numerical

Φ_p -optimal designs for some p, -∞<p≤1 are

obtained.Advisory Committee none - chair

none - co-chair

none - co-chair

none - co-chair

none - co-chair

Mong-Na Lo Huang - advisor

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etd-0205109-154131.pdf Date of Submission 2009-02-05