臺灣博碩士論文加值系統

在本論文中，我們討論異方差模型中大中取小的最適設計。此論文的目的為比較三個大中取小準則。首先，根據Wong(1998)的演算法，我們先求得針對這三個不同準則下的最適設計，並且比較它們的相對效率。最後，對於簡單線性模型，在一些簡單的條件下，我們證明這三個大中取小的最適設計是相等的。

In this thesis, we are interested in finding optimal minimax designs for heteroscedastic model. The goal of this thesis is to study three minimax criteria. Here for different efficiency functions, the corresponding numerically optimal minimax designs are computed according to the generating algorithm of Wong (1998), and then their relative efficiencies are compared. Finally for simple linear model, under some simple conditions, we show these three types of optimal minimax designs are equivalent.

1 Introduction 1
2 Equivalence theorem and generating algorithm for optimal minimax
designs 3
2.1 Equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Generating algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Optimal minimax designs for simple linear model 7
3.1 Numerically optimal minimax designs . . . . . . . . . . . . . . . . . . . . 8
3.2 Symmetric convex e±ciency function . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Symmetric interval . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 Extrapolation interval . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Quadratic and cubic models 18
4.1 Quadratic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Cubic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Conclusion 21
A Appendix 24
A.1 The Figures of ci(x; ¹¤; ¼¤), i = 1; 2; 3. . . . . . . . . . . . . . . . . . . . 24
A.1.1 The simple liner model with ¸(x) = x + 5. . . . . . . . . . . . . . 24
A.1.2 The simple liner model with ¸(x) = exp(¡5x2). . . . . . . . . . . 25
A.1.3 The simple liner model with ¸(x) = 0:5x2 + 1. . . . . . . . . . . . 26
A.1.4 The quadratic model with ¸(x) = 0:5x2 + 1. . . . . . . . . . . . . 27
A.1.5 The quadratic model with ¸(x) = exp(x2). . . . . . . . . . . . . . 28
A.1.6 The cubic model with ¸(x) = 0:5x2 + 1. . . . . . . . . . . . . . . 29
A.2 Proof of Theorem 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.3 Proof of Theorem 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References 32

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