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論文名稱 Title |
混合實驗在Scheffe'模型之穩健A-最適設計 Robust A-optimal designs for mixture experiments in Scheffe' models |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
43 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2003-06-06 |
繳交日期 Date of Submission |
2003-07-28 |
關鍵字 Keywords |
最適設計 equivalence theorem, invariant symmetric block matrices, robust design., Convex combination |
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統計 Statistics |
本論文已被瀏覽 5737 次,被下載 1762 次 The thesis/dissertation has been browsed 5737 times, has been downloaded 1762 times. |
中文摘要 |
在這文章中,我們對於混和實驗考慮在Scheffe(1958)的線性和二次模型中調查穩健A-最適設計. 在陳令由(2000),對於線性和二次模型的A-最適設計已經呈現,我們將利用在一些穩健的A最適方法的結果下,去發現一些線性和二次兩者的穩健設計.而這設計,也被證明會是線性和二次的最適設計的線性組合. |
Abstract |
A mixture experiment is an experiments in which the q-ingredients are nonnegative and subject to the simplex restriction on the (q-1)-dimentional probability simplex. In this work , we investigate the robust A-optimal designs for mixture experiments with uncertainty on the linear, quadratic models considered by Scheffe' (1958). In Chan (2000), a review on the optimal designs including A-optimal designs are presented for each of the Scheffe's linear and quadratic models. We will use these results to find the robust A-optimal design for the linear and quadratic models under some robust A-criteria. It is shown with the two types of robust A-criteria defined here, there exists a convex combination of the individual A-optimal designs for linear and quadratic models respectively to be robust A-optimal. In the end, we compare efficiencies of these optimal designs with respect to different A-criteria. |
目次 Table of Contents |
Contents 1. Introduction 2. Preliminary 3. The robust A-optimal designs for Scheffe's linear and quadratic models 4. Relation functions of r and a 5. Summary and further works |
參考文獻 References |
[1] Box, G. E. P., and Draper, N. R. (1959). A Basis for the Selection of a Response Surface Design, Journal of the American Statistical Association, 54, 622-654. [2] Chan, L. Y. (2000). Optimal Design for Experiments with Mixtures: A Survey. Communications in Statistions Instatistics-Theory and Methods, 29 (9 & 10), 2281-2312. [3] Guan, Y., Chao, X. (1987). On the A-optimal Allocation of Observations for the Generalized Simplex-Centroid Design (in Chinese). Journal of Engineering Mathematics, 4(3), 33-39. [4] Dette, H. (1990). A Generalization of D- and Ds- Optimal Design in Polynomial Regression. The Annals of Statistics. Vol. 18, No. 4, 1784-1804. [5] Draper, N. R., Gaffke, N. and Pukelsheim, F. (1993) Rotatability of Cariance Surfaces and Moment Matrices. Journal of Statistical Planning and Inference. 36, 347-356. [6] Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiffer Ordering of Simplex Designs for Second-Degree Mixture Models with four or more Ingredients. The Annals of Statistics, Vol. 28. No. 2, 578-590. [7] Draper, N. R., Pukelsheim, F. (1999). Kiffer Ordering of Simplex Designs for first- and second-degree Mixture Models. Journal of Statistical Planning and Inference 79, 325-348. [8] Draper, N. R., Pukelsheim, F. (1998). Mixture Models based on Homogeneous Polynomials. Journal of Statistical Planning and Inference 71, 303-311. [9] Fedorov V. V. (1972). Theory of Optimal Experiments. Translated and edited by W. J. Studden and E. M. Klimko. Academic press, New York. [10] Kiefer, J. (1961). Optimal Designs in Regression Problems, II. Annals of Mathematical Statistics, Vol. 32, Issue 1, 298-325. [11] Klein, T. (2002). Invariant Sysmmetric Block Matrices for the Design of Mixture Experiments. Institutfur Mathematik, Universitat Augsburg, Report No. 443. [12] Klein, T. (2002). Optimal Designs for Second-degree Kronecker Model Mixture Experiments. it Institutfur Mathematik, Universitat Augsburg, Report No. 444. [13] Pukelshiem, F. (1993). Optimal Design of Experiments. Wiley, New York. [14] Scheffe, H. (1958). Experiments with Mixtures. Journal of the Royal Statistical Society Series B 20, 344-360. |
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