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論文名稱 Title |
多項式迴歸模型具權重函數 exp(alpha x)之D 最適設計 D-optimal designs for polynomial regression with weight function exp(alpha x) |
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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 |
20 |
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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 |
2007-05-24 |
繳交日期 Date of Submission |
2007-06-25 |
關鍵字 Keywords |
Legendre 多項式、D 最適設計、Laguerre 多項式、arcsin 分布、D 等價定理、夾擠定理 Squeeze Theorem, Legendre polynomial, Laguerre polynomial, D-Equivalence Theorem, asymptotic design, arcsin distribution, D-optimal design |
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統計 Statistics |
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中文摘要 |
在這篇論文中,探討d 維的多項式迴歸模型具權重函數Exp(α x),而D 最適設計 ξ_d^* 可藉由三個微分方程的式子完全刻畫出來,且一些轉換不變的性質也有推導並証明。當d 趨近於無窮大時,我們也證明此D 最適設計ξ_d^* 會弱收歛至arcsin 分布,且此篇論文也做了ξ_d^* 與arcsin 分布的比較。 |
Abstract |
Weighted polynomial regression of degree d with weight function Exp(α x) on an interval is considered. The D-optimal designs ξ_d^* are completely characterized via three differential equations. Some invariant properties of ξ_d^* under affine transformation are derived. The design ξ_d^* as d goes to 1, is shown to converge weakly to the arcsin distribution. Comparisons of ξ_d^* with the arcsin distribution are also made. |
目次 Table of Contents |
Contents Abstract ....................................................................ii 1 Introduction ..........................................................1 2 Preliminary ...........................................................2 3 Asymptotic distribution of ξ_d^* .......................4 4 Examples ..............................................................7 5 Conclusions .........................................................11 Appendix ...................................................................12 References ...............................................................14 |
參考文獻 References |
References Chang, F.-C. and Lin, G.-C. (1997). D-optimal designs for weighted polynomial regres- sion. J. Statist. Plann. Inference 62, 317-331. Fedorov, V.V. (1972). Theory of Optimal Experiments (Translated by W.J. Studden and E.M. Klimko). Academic press, New York. Hoel, P.G. (1958). E±ciency problems in polynomial estimation. Ann. Math. Statist. 29, 1134-1145. Karlin, S. and Studden, W.J. (1966). Optimal experimental designs. Ann. Math. Statist. 37, 783-815. Kiefer, J.C. andWolfowitz, J. (1960). The equivalence of two extremum problems. Canad. J. Math. 12, 363-366. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2002). Numerical Recipes in C++, 2nd edition. Cambridge University Press, London. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. Silvey, S.D. (1980). Optimal Design. Chapman & Hall, London. Szeg |
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