本論文主要是探討在封閉區間中具有有界及恆正權重函數的d次單變數多項式迴歸模型的D最適設計。當d趨近於無限大時，我們證明D最適設計會弱收歛至反正弦分佈。對於權種函數等於1時，我們推導出五種設計其D準則值的公式，包含(一)均勻密度設計；(二)反正弦密度設計；(三)J_{1/2,1/2}密度設計；(四)反正弦支撐點設計；(五)均勻支撐點設計。除了探討這五種設計的D效率比較，也給出它們D效率的漸進展開式和極限值。從中我們證明前四種設計中，反正弦支撐點設計的D效率是最高的。

Consider the D-optimal designs for the dth-degree polynomial regression model with a bounded and positive weight function on a compact interval. As the degree of the model goes to infinity, we show that the D-optimal design converges weakly to the arcsin distribution. If the weight function is equal to 1, we derive the formulae of the values of the D-criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J_{1/2,1/2} density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs are investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.

Contents

Abstract . . . . . . . . . . . . .. . . . . . . . . . . . . . ii
1 Introduction . . . . . . . .. . . . . . . . . . . . . . 1
2 Preliminaries. . . . . .. . . . . . . . . . . . . . . 3
3 Arcsin limit theorem of ξ_d^* . . . . . . . 4
4 D-effciency . . . . . . . . . . .. . . . . . . . . . . . 5
5 Conclusions. . . . . . . . . . .. . . . . . . . . . . 8
Appendix . . . . . . . . . . . .. . . . . . . . . . . . . . 9
A.1 Proof of Lemma 2.2 . . . . . . . . . . . . . 9
A.2 Proof of Lemma 2.3 . . . . . . . . . . . . 10
A.3 Proof of Lemma 4.1 . . . . . . . . . . . . 10
A.4 Proof of Lemma 4.2 . . . . . . . . . . . . 12
A.5 Proof of Lemma 4.3 . . . . . . . . . . . . 14
A.6 Proof of Lemma 4.4 . . . . . . . . . . . . 14
A.7 Proof of Lemma 4.5 . . . . . . . . . . . . 15
References. . . . . . . . . . . .. . . . . . . . . . . . 20

List of Tables
1 Maximum point (α,β) of det M(ξ_d^*) with various d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 D-effciencies of ξ_{0,0}, ξ_{-1/2,-1/2}, ξ_{1/2,1/2}, σ_d, and λ_d for d = 1,2,..., 12 . . . . 23

List of Figures
1 D-effciencies of ξ_{α,β}, with 0.9≤t≤6 for d = 1, 2,..., 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 D-effciencies of ξ_{0,0}, ξ_{-1/2,-1/2}, ξ_{1/2,1/2}, σ_d, and λ_d for d = 1, 2,..., 50. . . . 25

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