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論文名稱 Title |
多反應值多項式模型之正合D型最適設計
Exact D-optimal designs for multiresponse polynomial model |
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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 |
14 |
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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 |
2000-05-26 |
繳交日期 Date of Submission |
2000-06-29 |
關鍵字 Keywords |
多反應變數、設計效率、D型最適設計、正合設計、多項式迴歸 multiresponse, efficiency of designs, exact design, polynomial regression, D-optimal design |
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統計 Statistics |
本論文已被瀏覽 5754 次,被下載 1974 次 The thesis/dissertation has been browsed 5754 times, has been downloaded 1974 times. |
中文摘要 |
考慮有單一控制變數與任意反應值共變異矩陣的多反應值多項式迴歸模型。Krafft、Schaefer(1992)與Imhof(2000)已求出,包含一個一次多項式反應值和一個二次多項式反應值之多反應值迴歸模型的n點D型最適設計。在本文中,將對前述結果,做出更進一步的探討。首先,我們可證明,D型正合最適設計在對控制變數作線性變換之中,具有不變性。此外,我們將會給出,只包含一次和二次多項式,且取值在[-1,1]區間的反應值之正合D型最適設計,並包括這個模型的正合D型最適設計對二次多項式單變數迴歸模型的正合D型最適設計之效率。最後,也將給出數值計算上的猜想結果。 |
Abstract |
Consider the multiresponse polynomial regression model with one control variable and arbitrary covariance matrix among responses. The present results complement solutions by Krafft and Schaefer (1992) and Imhof (2000), who obtained the n-point D-optimal designs for the multiresponse regression model with one linear and one quadratic. We will show that the D-optimal design is invariant under linear transformation of the control variable. Moreover, the most cases of the exact D-optimal designs on [-1,1] for responses consisting of linear and quadratic polynomials only are derived. The efficiency of the exact D-optimal designs for the univariate quadratic model to that for the above model are also discussed. Some conjectures based on intensively numerical results are also included. |
目次 Table of Contents |
1. Introduction 2. Preliminaries 3. Main results 4. Efficiency and conjectures 5. Discussion |
參考文獻 References |
1. Atkins, J. E. and Cheng, C. S. (1999). Optimal regression designs in the presence of random block effects. J. Statist. Plann. Infer., 77, 321-335. 2. Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs. Oxford University Press, New York. 3. Bischoff, W. (1993). On D-optimal designs for linear models under correlated observations with an application to a linear model with multiple response. J. Statist. Plann. Infer., 37, 69-80. 4. Bischoff, W. (1995). Determinant formulas with applications to designing when the observations are correlated. Ann. Inst. Statist. Math., 47, 385-399. 5. Chang, F.-C. and Yeh, Y.-R. (1998). Exact A-optimal designs for quadratic regression. Statist. Sinica, 8, 527-533. 6. Cheng, C. S. (1995). Optimal regression designs under random block-effects models. Statist. Sinica, 5, 485-497. 7. Dette, H. (1990). A generalization of D- and D1-optimal designs in polynomial regression. Ann. Statist., 18, 1784-1804. 8. Draper, N. R. and Hunter, W. G. (1966). Design of experiments for parameter estimation in multiresponse situations. Biometrika, 53, 525-533. 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. Gaffke, N. and Krafft, O. (1982). Exact D-optimum designs for quadratic regression. J. Roy. Statist. Soc. Ser. B, 44, 394-397. 11. Imhof, L. A. (2000). Optimal designs for a multiresponse regression model. J. Multivar. Anal., 72, 120-131. 12. Kim, W. B. and Draper, N. R. (1994). Choosing a design for straight line fits to two correlated responses. Statist. Sinica, 4, 275-280. 13. Krafft, O. and Schaefer, M. (1992). D-optimal designs for a multivariate regression model. J. Multivar. Anal., 42, 130-140. 14. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. 15. Silvey, S. D. (1980). Optimal Design. Chapman & Hall, London. |
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