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論文名稱 Title |
無截距項多項式迴歸模型之c-最適設計
C-optimal designs for polynomial regression without intercept. |
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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 |
31 |
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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 |
2002-06-07 |
繳交日期 Date of Submission |
2003-02-17 |
關鍵字 Keywords |
c-最適設計 individual regression coecient., Elfving Theorem, c-optimal design |
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統計 Statistics |
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中文摘要 |
在本論文中,我們討論無截距項多項式迴歸模型之c-最適設計。Huang and Chen 於1996年已證出在 $[-1,1]$估計某些個別迴歸係數時,有截距項 d階多項式迴歸模型之c-最適設計與無截距項相同。我們找到 在[-1,1]估計其他未被證出的個別迴歸迴歸係數之c-最適設計。針對無截距項模型,我們證出支撐點(support points) 在 [-b,b]具尺度不變性(scale invariant)。最後我們利用 Elfving定理討論在不對稱區間中估計 2階無截距項多項式迴歸模型之迴歸係數。 |
Abstract |
In this work, we investigate c-optimal design for polynomial regression model without intercept. Huang and Chen (1996) showed that the c-optimal design for the dth degree polynomial with intercept is still the optimal design for the no-intercept model for estimating certain individual coe cients over [−1, 1]. We found the c-optimal designs explicitly for estimating other individual coe cients over [−1, 1], which have not been obtained earlier. For the no-intercept model, it is shown that the support points are scale invariant over [−b, b]. Finally some special cases are discussed for estimating the coe cients of the 2nd degree polynomial without intercept by Elfving theorem over nonsymmetric interval [a, b]. |
目次 Table of Contents |
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 1 Introduction . . . . . . . . . .. . . . . . . . . . . . . . . 1 2 Preliminary for characterization of c-optimal designs and related results. . . . . . . . . . . . . 3 3 Optimal designs for the individual regression coecients . . . . . . . . . . . . . 7 3.1 Individual regression coeffcients for the no-intercept model over [-1,1] 7 3.2 Individual regression coeffcients for the no-intercept model over [-b,b] 15 3.3 Individual regression coeffcients for the no-intercept model over [a,b] 17 4 Discussion . . . . . . . . . . . . . . .. . . . . 21 References . . . . . . . . . . . . . . . . . .. . . 22 |
參考文獻 References |
Chang, F.C. and Heiligers, B. (1996) E-optimal designs for polynomial regression without intercept. {J. Statist. Plann. Inference.} {55}, 371-387. Elfving, G. (1952) Optimum allocation in linear regression theory. {Ann. Math. Statist.} {23}, 255-262. Fedorov V. V. (1972) {Theory of Optimal Experiments.} Translated and edited by W. J. Studden and E.M. Klimko. Academic press, New York. Graybill, F. A. (1983) {Matrices with Applications in Statistics.} Wadsworth, Belmont, California. Hoel, P. G. and Levine, A. (1964) Optimal spacing and weighting in polynomial prediction. {Ann. Math. Statist.} {35}, 1553-1560. Huang, M.-N. L., Chang, F.C. and Wong, W.K. (1995) D-optimal designs for polynomial regression without an intercept. { Statistica Sinica.} {5}, 441-458. Huang, M.-N. L. and Chen, R.B. (1996) $C$-optimal designs for regression models with weak chebyshev property. Technical Report, Department of Applied Mathematics National Sun Yat-sen University. Kifer, J. and Wolfowitz, J. (1965) On a theorem of Hoel and Levine on extrapolation. {Ann. Math. Statist.} {36}, 1627-1655. Kitsos, C.P., Titterington, D. M. and Torsney, B. (1988) An optimal design problem in rhymometry. {Biometrics} {44}, 657-671. Pukelsheim, F. and Torsney, B. (1991) Optimal weights for experimental designs on linearly independent support points. {Ann. Statist.} {19}, 1614-1625. Pukelsheim, F. (1993) {Optimal Design of Experiments.} Wiley, New York. Studden, W.J. (1971) Elfving's Theorem and optimal designs for quadratic loss. {Ann. Math. Statist.} {42}, 1613-1621. Studden, W.J. (1968) Optimal designs on Tchebycheff points. {Ann. Math. Statist.} {39}, 1435-1447. |
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