A Note on Generalized Estimation in Response Surfaces

廣義估計量在反應曲面中的注解

黃國男(Kuo-Nan Huang)

《品質學報》 15卷6期 (2008/12) Pp. 441-450

摘要

在本論文中，將介紹如何使用廣義估計量(GE)法來求解無曲率模式的最佳二階設計動差。同時，針對反應曲面法(RSM)文獻中，有關Box and Draper (1959, 1963) 所提之平均均方誤差準則(AMSE)下的單變數設計問題，也將呈現以廣義估計量法與最小平方估計量(LSE)法所求解後的比較結果。在這個特定的問題中，發現了最小平方估計量法下的AMSE值將視設計配置與實際曲率而定。然而，廣義估計量法下的AMSE值將依參數權重、設計配置、在原始模式中的參數與實際曲率而定。為了公平比較在這兩個方法下的AMSE值，首先給定一個固定的曲率，則不同的相對曲率比將被指定用來定義存在假設模式中的尺度參數。藉由表1的比較結果可知，當曲率呈現顯著時，廣義估計量法在整體上執行的比最小平方估計量法好。因此，這個研究除了廣義估計量法的技術注解外，也將提供這兩個方法在標準反應曲面法下的比較結果。而這些比較報告也可以作為導引反應曲面實務專家針對在廣義估計量法下，如何選擇一個合適的設計點來達到較小的AMSE值。

關鍵字

廣義估計量GE ； 反應曲面法RSM ； 平均均方誤差準則AMSE ； 最小平方估計量LSE

並列摘要

In this article, a practical assumption on the neglected curvature is introduced for computing the optimum second-order design moment for the generalized estimation (GE) approach (Kupper and Myedrech, 1973, 1974). We also present the comparison results of the GE approach to the least-squares estimation (LSE) approach within the context of Box and Draper's (1959, 1963) average mean squared error (AMSE) criterion specifically for a single variable design problem appearing in the RSM literature. For this particular problem, it has been discovered that the AMSE of the LSE approach depends only on the design allocation and the actual scaled curvature. Nonetheless, the AMSE of the GE approach relies on the parameter weight, the design allocation, the actual scaled parameter existing in the initial model, and the actual scaled curvature. To make fair comparison possible, a fixed scaled curvature is first given for LSE and GE, and then a variety of relative curvature ratios are specified to define the scaled parameter already exiting in the assumed model. The comparison results from Table 1 show that the GE approach, on the whole, performs better than the LSE approach in the AMSE value for the problem discussed in this paper. The improvement of the GE approach over the LSE approach becomes more prominent as the potential bias error is significantly present. In additional to being a technical note on the GE approach, this research supplies supplementary comparison results between GE and LSE for standard RSM textbooks. The comparison reports can also serve as a practical guide for RSM practitioners to choose a suitable design point arrangement for the GE approach to achieve minimum AMSE.

並列關鍵字

generalized estimation GE ； response surface methodology RSM ； average mean squared error ASME ； least squares estimation LSE

參考文獻

Box, G. E. P.,Draper, N. R.(1959).A basis for the selection of a response surface design.Journal of the American Statistical Association.54,622-654.

Box, G. E. P.,Draper, N. R.(1963).The choice of a second order rotatable design.Biometrika.50(3),335-352.

Box, G. E. P.,Draper, N. R.(1987).Empirical Model-Building and Response Surfaces.New York:John Wiley & Sons.

Draper, N. R.,Lawrence W. E.(1965).Designs which minimize model inadequacies: cuboidal regions of interest.Biometrika.52,111-118.

Huang, K. N.,Fan, S. K.(2004).A note on minimum bias estimation in response surfaces.Statistics and Probability Letters.70(1),71-85.