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研究生:陳盈螢
研究生(外文):Ying-Ying Chen
論文名稱:無截距項多項式迴歸模型之c-最適設計
論文名稱(外文):C-optimal designs for polynomial regression without intercept.
指導教授:羅夢娜羅夢娜引用關係
指導教授(外文):Mong-Na Lo Huang
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:英文
論文頁數:31
中文關鍵詞:c-最適設計
外文關鍵詞:individual regression coe cient.Elfving Theoremc-optimal design
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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。
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].
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 Introduction . . . . . . . . . .. . . . . . . . . . . . . . . 1
2 Preliminary for characterization of c-optimal designs and related results.
. . . . . . . . . . . . 3
3 Optimal designs for the individual regression coe cients . . . . . . . . . . . . . 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
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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