我們將要探討的是有關加權多項式迴歸模型D之最適設計的問題，在近代文章中已經證明加權多項式迴歸模型之設計架構問題，相等於多項式迴歸模型伴隨著加權函數在[-1,1]區間之設計架構問題
藉由泰勒展開式此篇論文提供一個遞迴演算法使得加權多項式迴歸模型D之最適設計的問題可以在有興趣的範圍被正確的估計。

This paper is concerned with the problem of computing approximate
D-optimal design for polynomial regression with analytic weight
function on the interval [m_0-a,m_0+a]. It is shown that the
problem of constructing D-optimal designs for weighted
polynomial regression is equivalent to that of finding D-optimal designs for polynomial regression with weight function |x|^s on the interval [-1,1] as a to 0. By using
Taylor expansion we provide a recursive algorithm such that the
D-optimal designs for weighted polynomial regression on a
neighborhood of m_0 can be determined.

Abstract.............3
Introduction...........4
Preliminary results........ 5
D-optimal designs............8
Example.............10
Conclusions.............11
References.......12
Appendix......14

Antille, G., Dette, H., Weinberg, A. (2003).
A note on optimal designs in weighted polynomial regression
for the classical efficiency functions.
{it J. Statist. Plann. Inference} {f 113}, 285-292.


Chang, F.-C. (1998).
On asymptotic distribution of optimal design for polynomial-type
regression.
{it Statist. and Probab. Letters} {f 36}, 421-425.


Chang, F.-C. and Lin, G.-C. (1997).
$D$-optimal designs for weighted polynomial regression.
{it J. Statist. Plann. Inference} {f 55}, 371-387.


Dette, H. (1990).
A generalization of $D$- and $D_1$-optimal designs in polynomial
regression.
{it Ann. Statist.} {f 18}, 1784-1804.


Dette, H. (1992).
Optimal designs for a class of polynomials of odd or even
degree.
{it Ann. Statist.} {f 20}, 238-259.


Dette, H., Haines, L.M. and Imhof, L. (1999).
Optimal designs for rational models and weighted polynomial
regression.
{it Ann. Statist.} {f 27}, 1272-1293.


Dette, H., Melas, V.B. and Biedermann, S. (2002).
A functional-algebraic determination of $D$-optimal designs
for trigonometric regression models on a partial circle.
{it Statist. and Probab. Letters} {f 58}, 389-397.

Dette, H., Melas, V.B. and Pepelyshev, A. (2000).
Optimal designs for estimating individual coefficients--a functional approach,
Preprint, Ruhr-Universit"at Bochum.
http://www.ruhruni-bochum.de/mathematik3/preprint.htm


Dette, H., Melas, V.B. and Pepelyshev, A. (2002).
$D$-optimal designs for trigonometric regression models on a partial
circle.
{it Ann. Inst. Statist. Math.} {f 54}, 945-959.



Fedorov, V.V. (1972).
{it Theory of Optimal Experiments}.
Translated and edited by W.J. Studden and E.M. Klimko.
Academic Press, New York.




Hoel, P. G. (1958).
Efficiency problems in polynomial estimation.
{it Ann. Math. Statist.} {f 29}, 1134-1145.


Huang, M.-N.L., Chang, F.-C. and Wong, W.-K. (1995).
$D$-optimal designs for polynomial regression without an
intercept.
{it Statist. Sinica} {f 5}, 441-458.


Imhof, L., Krafft, O. and Schaefer, M. (1998).
$D$-optimal designs for polynomial regression with weight function
$x/(1+x)$.
{it Statist. Sinica} {f 8}, 1271-1274.


Karlin, S. and Studden, W.J. (1966).
Optimal experimental designs.
{it Ann. Math. Statist.} {f 37}, 783-815.

Karlin, S. and Studden, W.J. (1966).
{it Tchebycheff System: With Applications in Analysis and
Statistics}.
Wiley, New York.


Khuri, A.I. (2002).
{it Advanced Calculus with Applications in Statistics}. 2nd edition.
Wiley, New York.


Kiefer, J. C. and Wolfowitz, J. (1960).
The equivalence of two extremum problems.
{it Canad. J. Math.} {f 12}, 363-366.


Lau, T.S. and Studden, W.J. (1985).
Optimal designs for trigonometric and polynomial regression
using canonical moments.
{it Ann. Statist.} {f 13}, 383-394.


Silvey, S.D. (1980).
{it Optimal Design}.
Chapman & Hall, London.



Studden, W.J. (1982).
Optimal designs for weighted polynomial regression using
canonical moments.
In {it Statistical Decision Theory and Related Topics III}
(ed. S.S. Gupta and J.O. Berger),
335-350, Academic Press, New York.


Szeg"o, G. (1975).
{it Orthogonal Polynomials},
4th edition. American Mathematical Society Colloquium Publications
{f 23}, Providence, Rhode Island.


Wolfram, S. (1999).
{it The Mathematica Book}. 4th edition.
Wolfram Media/Cambridge University Press.