這篇文章主要是研究在區間 [m-a, m+a]，具有可解析加權函式多項式迴歸的A-最適設問題。當a是趨近於0時，此最適設計的結構僅決於a和加權函式。此外，假如此加權函式在a=0時是可解析函式，則一個伸縮的最適設計點和權重a=0時是可解析函式。利用泰勒展開式係數可以決定一個遞迴公式來計算A-最適設計。

This paper is concerned with the problem of constructing
A-optimal design for polynomial regression with analytic weight
function on the interval [m-a,m+a]. It is
shown that the structure of the optimal design depends on a and
weight function only, as a close to 0. Moreover, if the weight
function is an analytic function a, then a scaled version of
optimal support points and weights is analytic functions of a at
$a=0$. We make use of a Taylor expansion which coefficients can be
determined recursively, for calculating the A-optimal designs.

Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . ii
1. Introduction . . . . . . . . . . . . . . . . . . . 1
2. Preliminary . . .. . . . . . . . . . . . . . . . . 2
3. Taylor expansion for A-optimal designs . . . . . . . . . . . . . . . . . . . . . . . 5
4. Example . . . . . . . . . . . . . . . . . . . . . 7
5. Conclusions . . . . . . .. . . . . . . . . . . . . 8
Appendix . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . 12

References
Acton, F.S. (1990). Numerical Methods that Work. The Mathematical Association of
America, Washington, D.C.
Antille, G., Dette, H. and Weinberg, A. (2003). A note on optimal designs in weighted
polynomial regression for the classical e±ciency functions. J. Statist. Plann. Infer-
ence 113, 285-292.
Chang, F.-C. and Heiligers, B. (1996). E-optimal designs for polynomial regression with-
out intercept. J. Statist. Plann. Inference 55, 371-387.
Chang, F.-C. and Lin, G.-C. (1997). D-optimal designs for weighted polynomial regres-
sion. J. Statist. Plann. Inference 62, 317-331.
Dette, H. (1993). A note on E-optimal designs for weighted polynomial regression. Ann.
Statist. 21, 767-771.
Dette, H., Haines, L.M. and Imhof, L. (1999). Optimal designs for rational models and
weighted polynomial regression. Ann. Statist. 27, 1272-1293.
Dette, H., Melas, V.B. and Pepelyshev, A. (2004). Optimal designs for estimating indi-
vidual coe±cients in polynomial regression{a functional approach. J. Statist. Plann.
Inference 118, 201-219.
Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments with Applica-
tions in Statistics, Probability, and Analysis. Wiley, New York.
Fedorov, V.V. (1972). Theory of Optimal Experiments. Translated and edited by Studden,
W.J. and Klimko, E.M. Academic Press, New York.
Heiligers, B. (1994). E-optimal designs in weighted polynomial regression. Ann. Statist.
22, 917-929.
Hoel, P.G. (1958). E±ciency problems in polynomial estimation. Ann. Math. Statist.
29, 1134-1145.
12
Huang, M.-N.L., Chang, F.-C. and Wong, W.-K. (1995). D-optimal designs for polyno-
mial regression without an intercept. Statist. Sinica 5, 441-458.
Huda, S. and Mukerjee, R. (1991). A-optimal design measures for polynomial regression.
Pakistan J. Statist. 7, 47{52.
Imhof, L., Kra®t, O. and Schaefer, M. (1998). D-optimal designs for polynomial regression
with weight function x=(1 + x). Statist. Sinica 8, 1271-1274.
Karlin, S. and Studden, W.J. (1966a). Optimal experimental designs. Ann. Math.
Statist. 37, 783-815.
Karlin, S. and Studden, W.J. (1966b). Tchebyche® System: With Applications in Analysis
and Statistics. Wiley, New York.
Kiefer, J. C. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann.
Math. Statist. 30, 271-294.
Kiefer, J.C. andWolfowitz, J. (1960). The equivalence of two extremum problems. Canad.
J. Math. 12, 363-366.
Khuri, A.I. (2003). Advanced Calculus with Applications in Statistics. 2nd edition. Wiley,
New York.
Lau, T.S. and Studden, W.J. (1985). Optimal designs for trigonometric and polynomial
regression using canonical moments. Ann. Statist. 13, 383-394.
Melas, V.B. (1978). Optimal designs for exponential regression. Math. Oper. Forsch.
Statist. Ser. Statist. 9, 45-59.
Melas, V.B. (2000). Analytic theory of E-optimal designs for polynomial regression. In:
Balakrishnan, N. (Ed.), Advances in Stochastic Simulation Methods, 85-116. Springer
Verlag, New York.
Montgomery, D.C., Peck, E.A. and Vining, G.G. (2001). Introduction to Linear Regression
Analysis, 3rd edition. Wiley, New York.
Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
Pukelsheim, F. and Studden, W.J. (1993). E-optimal designs for polynomial regression.
Ann. Statist. 21, 402-415.
Pukelsheim, F. and Torsney, B. (1991). Optimal weights for experimental designs on
linearly independent support points. Ann. Statist. 19, 1614-1625.
Rivlin, T. J. (1990). Chebyshev Polynomials. Wiley, New York.
13
Silvey, S.D. (1980). Optimal Design. Chapman & Hall, London.
Studden, W.J. (1982). Optimal designs for weighted polynomial regression using canonical
moments. In Statistical Decision Theory and Related Topics III (ed. S.S. Gupta and
J.O. Berger), 335-350. Academic Press, New York.
Studden, W.J. and Tsay, J.Y. (1976). Remez's procedure for ‾nding optimal designs.
Ann. Statist. 4, 1271-1279.
Wolfram, S. (2003). The Mathematica Book, 5th edition. Wolfram Media, Champaign,
IL.