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。

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.

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