| 摘要(英) |
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. |
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