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系統識別號 U0026-0812200910353506
論文名稱(中文) 模型區別與參數估計的準則穩健最適設計之研究
論文名稱(英文) A Study on Criterion-Robust Optimal Designs for Model Discrimination and Parameter Estimation
校院名稱 成功大學
系所名稱(中) 統計學系碩博士班
系所名稱(英) Department of Statistics
學年度 91
學期 2
出版年 92
研究生(中文) 蔡旻曉
研究生(英文) Min-Hsiao Tsai
電子信箱 mhtsai28@yahoo.com.tw
學號 r2887101
學位類別 博士
語文別 英文
論文頁數 107頁
口試委員 口試委員-劉應興
口試委員-呂金河
口試委員-蔡風順
口試委員-羅夢娜
指導教授-任眉眉
中文關鍵字 有效性  多重目的最適性準則  傅立葉迴歸模型  多項式迴歸模型  Mr*-最適設計  小中取大原則  正規動差  Mr-最適性準則 
英文關鍵字 efficiency  canonical moment  Fourier regression model  Mr-optimality criterion  polynomial regression model  multiple-objective  maximin principle 
學科別分類
中文摘要 本研究主要探討兩個在[-1,1]^q上的多項式迴歸模型
(polynomial regression models)或兩個在[-π,π]上的傅
立葉迴歸模型 (Fourier regression models) 之區別問題,
與以往不同的是,在此我們同時將各個模型上有關參數估計
的問題考量進來。為了獲得能有效兼顧上述模型區別與參數
估計兩目的之實驗設計,在本研究中我們提出了一個多重目
的最適性準則(multiple-objective optimality criterion)
稱之Mr-最適性準則 (Mr-optimality criterion)。簡單來
講,此一準則為Ds-有效性(用以評估一實驗設計在模型區別
表現上優劣與否的指標)及D-有效性(用以評估一實驗設計在
參數估計表現上優劣與否的指標) 的加權幾何平均數,其主
要特色是除了給予r(0≦r≦1)的權重於模型區別目的外,另
一方面亦對參數估計給予(1-r)的權重保障。在此Mr-最適性
準則下,我們利用正規動差(canonical moments) 的技巧成
功地解得其所對應的Mr-最適設計(Mr-optimal design)的解
析解。此外,我們亦研究了在不同的加權選取準則下,不同
的Mr'-最適設計在各種Mr-最適性準則下有效性的表現行為。
更進一步地,我們應用了小中取大原則(maximin principle)
發現 Mr'-最適設計的最小Mr-有效性值為r'的一個先增後降
函數,且這些最小Mr-有效性值的最大值發生於r'=r*,此意
味著所對應的Mr*-最適設計不管在任何 Mr-最適性準則下皆
會有不錯的表現。最後,數值分析的結果亦支持此一Mr*-最
適設計無論在多重目的考量或單一個別目的考量下均有相當
穩健的表現。

英文摘要 Consider the problem of discriminating between two
rival polynomial regression models on the q-cube
[-1,1]^q, qεN, or two rival Fourier regression
models on the circle [-π,π], and estimating
parameters in the models. In order to find
experimental designs which are efficient for both
purposes of model discrimination and parameter
estimation simultaneously, we propose a general
multiple-objective optimality criterion,
Mr-optimality criterion, which is a weighted
geometric average of D- and Ds-efficiencies, it
puts weight r (0≦r≦1) for model discrimination
and (1-r) for parameter estimation.
The corresponding Mr-optimal design is explicitly
derived in terms of canonical moments. Moreover,
the behavior of the proposed Mr-optimal designs
is investigated under different weighted selection
criterion. Furthermore, applying the maximin
principle on the efficiencies of experimental
designs, the extreme value of the minimum
Mr-efficiency of any Mr'-optimal design is obtained
at r'=r*, which results in the corresponding
Mr*-optimal design to be served as a criterion-
robust optimal design for the described problem.

論文目次 中文摘要--------------------------------------------i
Abstract-------------------------------------------ii
Acknowledgements----------------------------------iii
Contents-------------------------------------------iv
List of Tables------------------------------------vii
List of Figures----------------------------------viii

1 Introduction--------------------------------------1

2 Canonical Moments---------------------------------7

3 Univariate Polynomial Regression Models: PART I--10
  3.1 Introduction-----------------------------------10
  3.2 Construction of Mr-Optimal Design--------------14
  3.3 Criterion-Robust Optimal Design----------------16
    3.3.1 Efficiency of Mr-Optimal Design--------------16
    3.3.2 Minimum Mr-Efficiency of Mr'-Optimal Design--17
  3.4 Comparison with Some Special Designs-----------24
  3.5 Concluding Remarks-----------------------------42

4 Univariate Polynomial Regression Models: PART I--43
  4.1 Introduction-----------------------------------43
  4.2 Construction of Mr-Optimal Design--------------46
  4.3 Criterion-Robust Optimal Design----------------47
    4.3.1 Efficiency of Mr-Optimal Design--------------47
    4.3.2 Minimum Mr-Efficiency of Mr'-Optimal Design--48
  4.4 Comparison with Some Special Designs-----------53
  4.5 Concluding Remarks-----------------------------59

5 Multivariate Polynomial Regression Models--------60
  5.1 Introduction-----------------------------------60
  5.2 Construction of Mr-Optimal Product Design------64
  5.3 Criterion-Robust Optimal Product Design--------66
    5.3.1 Efficiency of Mr-Optimal Product Design------66
    5.3.2 Minimum Mr-Efficiency of Mr'-Optimal Product Design-------67
  5.4 Comparison with Some Special Designs-----------74
  5.5 Concluding Remarks-----------------------------79

6 Fourier Regression Models------------------------81
  6.1 Introduction-----------------------------------81
  6.2 Projection Design and Optimality Criteria for Fourier Regression Models-----84
  6.3 Construction of Mr-Optimal Design--------------86
  6.4 Criterion-Robust Optimal Design----------------91
    6.4.1 Efficiency of Mr-Optimal Design--------------91
    6.4.2 Minimum Mr-Efficiency of Mr'-Optimal Design--92
  6.5 Comparison with Some Special Designs-----------96
  6.6 Concluding Remarks----------------------------100

7 Conclusions-------------------------------------101

Bibliography--------------------------------------103
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