臺灣博碩士論文加值系統

本論文主要處理在區間[-1, 1]上有關k-1次多項式迴歸模型(模型A)與k次多項式迴歸模型(模型B)針對參數估計時之 Läuter-type 最適設計問題,主要的目的在尋求一個好的實驗設計可以同時有效地滿足二個目的:(1)假若模型A合適,則對模型A下的參數進行有效率的統計推論;(2)假若模型B合適, 則利用同一實驗設計對該模型B下的參數進行有效率的統計推論。針對上述考量,我們提出一個多重目的最適性準則(multiple-objective optimality criterion),稱為Ψα-最適性準則,並利用正規動差(canonical moments)技巧求得在Ψα-最適性準則下之最適設計的解析解。更進一步地,我們利用小中取大原則(maximin principle)建構一個對個別模型成立下有關參數估計問題均具穩健性質的實驗設計。最後,透過數值分析方法比較本研究中所提之最適設計以及兩個單一目的最適設計在不同多項式次冪考量下有效性的表現行為,整體而言,本論文所提的最適設計無論在多重目的或個別單一目的下均有相當不錯的表現。

Consider the problem of Läuter-type optimal design for the polynomial regression models with degree k-1 (Model A) and degree k (Model B) on the interval [-1, 1]. The purpose of this thesis is to seek experimental designs that can efficiently serve two objectives simultaneously : (1) to make inferences in Model A and (2) to make inferences in Model B. For this dual-objective consideration, we propose a multiple-objective optimality criterion－Ψα-optimality criterion, and the correspondingΨα-optimal design is derived in terms of canonical moments. Furthermore, by the aid of maximin principle, we construct a model-robust optimal design for the consideration of parameter estimation. Finally, we make a comparison of our proposed design with other traditional optimal designs for different degree k. The numerical results indicate that our proposed design is very robust for the described problem.

目錄
1 緒論 1
2 D-最適性準則與Ψα-最適性準則 3
3 模型穩健最適設計 5
3.1 正規動差與Ψα-最適設計 5
3.2 Ψα-最適設計之有效性 9
3.3 Ψα'-最適設計之最小Ψα-有效性 11
4 最適設計有效性之探討 17
5 結論 20
參考文獻 21
附錄I 24
附錄II 27

圖目錄
3.1 當k = 3時,Ψα'-最適設計的Ψα-有效性之等高線圖 10
3.2 當k = 3時, min0≦α≦1{Ψα-eff(ηα')}與α'的關係圖 14
4.1 設計ηα* 、η1/2 、ηA及ηB 的min0≦α≦1{Ψα}-、 Ψα*- 、DA-及DB-有效性 19
II.1 極限設計η˜的分配示意圖 27
II.2 當k = 2時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 28
II.3 當k = 3時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 29
II.4 當k = 4時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 30
II.5 當k = 5時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 31
II.6 當k = 6時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 32
II.7 當k = 7時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 33
II.8 當k = 8時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 34
II.9 當k = 9時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 35
II.10 當k = 10時, 設計ηα* 、η1/2 、ηA及ηB 的支撐點與權重之直方示意圖 36

表目錄
3.1 不同k之下的α*與最小Ψα-有效性值 15
3.2 設計ηα* 和η1/2之支撐點及權重 16
4.1 當k = 3時, 一些設計之各種有效性的比較 18

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