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博碩士論文 etd-0710106-042830 詳細資訊
Title page for etd-0710106-042830
論文名稱
Title
多變量迴歸模型之最適校準設計
Optimal Designs for Calibrations in Multivariate Regression Models
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
93
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2006-06-28
繳交日期
Date of Submission
2006-07-10
關鍵字
Keywords
古典估計量、最適校準設計、校準點、多變量迴歸模型、目標值
Maximin efficient design, Optimal calibration design, Relative potency, Scalar optimal design, Minimax design, Multivariate calibration, Location-shift parameter, Bioassay, C-criterion, Classical estimator, Equivalence theorem, Locally optimal design
統計
Statistics
本論文已被瀏覽 5722 次,被下載 1378
The thesis/dissertation has been browsed 5722 times, has been downloaded 1378 times.
中文摘要
本篇論文主要研究多反應變數迴歸模型在校準問題上的最適設計。眾所周知的校準問題是由一已知的反應值(或稱目標值)推論與此反應值相對應的未知控制變數的控制量,我們稱此控制量為校準點。一常見的作法是利用控制變數與反應變數的迴歸函數反求校準點。以此種方法求得的校準點的估計量稱為古典估計量。文獻上,已有許多討論校準問題的論文,但是與最適設計相關的論文卻相對較少,且都僅止於討論單反應變數的最適校準設計。在這篇論文裡,我們主要考慮的模型為一具有一個控制變數,但同時有多個具相關性的反應變數的線性迴歸模型。我們的主要目的是對一組給定的反應變數的目標值反求控制量的預測值時,可得到較佳估計校準點的最適校準設計。由於要達到各目標值的校準點可能互異,因此我們考慮的最適校準點為滿足最小化反應期望值與目標值差異的加權平方和的校準點。為了得到一個能準確的預測此控制量校準點的有效設計,我們選取最適校準設計的準則為能最小化校準點與它的估計量的差異的均方平均值的設計。在這個準則下,我們提出具有雙反應變數的簡單線性迴歸模型及二次迴歸模型的最適校準設計。
Abstract
In this dissertation we first consider a parallel linear model with correlated dual responses on a symmetric compact design region and construct locally optimal designs for estimating the location-shift parameter. These locally optimal designs are variant under linear
transformation of the design space and depend on the correlation between the dual responses in an interesting and sensitive way.

Subsequently, minimax and maximin efficient designs for estimating the location-shift parameter are derived. A comparison of the behavior of efficiencies between the minimax and maximin efficient designs relative to locally optimal designs is also provided. Both minimax or maximin efficient designs have advantage in terms of estimating efficiencies in different situations.

Thirdly, we consider a linear regression model with a
one-dimensional control variable x and an m-dimensional response variable y=(y_1,...,y_m). The components of y are correlated with a known covariance matrix. The calibration problem discussed here is based on the assumed regression model. It is of interest to obtain a suitable estimation of the corresponding x for a given target T=(T_1,...,T_m) on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of calibration point x, say x_0,
is considered to be the one which minimizes the weighted sum of squares of such deviations within the range of x. The objective of this study is to find a locally optimal design for estimating x_0, which minimizes the mean square error of the difference between x_0 and its estimator. It shows the optimality criterion is
approximately equivalent to a c-criterion under certain conditions and explicit solutions with dual responses under linear and quadratic polynomial regressions are obtained.
目次 Table of Contents
Abstract {i}
Acknowledgments {iii}
Contents {iv}
List of Tables {vii}
List of Figures {ix}
Chapters
1 Introduction {1}
2 Optimal Designs for Estimating the Location-shift Parameter of Parallel Models with Correlated Responses {6}
2.1 Introduction {7}
2.2 Location-shift parameter {9}
2.3 An application and discussions {18}
2.4 Appendix {20}
2.4.1 Proof of Theorem 2.2.5 {20}
3 Minimax and Maximin Efficient Designs for Estimating the Location-shift Parameter of Parallel Models with Dual Response {24}
3.1 Introduction {25}
3.2 Preliminaries {27}
3.3 Minimax designs {31}
3.4 Maximin efficient designs {33}
3.5 Discussions {35}
3.6 Appendix {39}
3.6.1 Proof of Theorem 3.3.4 {39}
3.6.2 Proof of Theorem 3.4.1 {42}
4 Optimal Designs for Calibrations in Multiresponse-univariate Regression Models {45}
4.1 Introduction {46}
4.2 Scalar optimal design for multiresponse linear regression model {49}
4.2.1 Preliminaries {49}
4.2.2 Scalar optimal design {51}
4.3 Optimal designs for calibrations {53}
4.3.1 Simple linear regression model {55}
4.3.2 Quadratic regression model {57}
4.4 An example {62}
4.5 Discussions {69}
4.6 Appendix {71}
4.6.1 The optimal control value for q_1,q_2 not in X {71}
4.6.2 The optimal control value for q_1 not in X and q_2 in X {72}
4.6.3 The coefficient vector {73}
References {74}
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