AMSE optimal design using generalized estimation for multi‐factor response surfaces

Abstract

AbstractWithin the realm of average mean squared error (AMSE), generalized estimation (GE; Kupper and Meydrech 1,2) is an attractive alternative to estimate the response surface. GE is based on the criterion that minimizes the variance of the fitted response $(\hat Y({\bf x}))$ by weighting the parameter estimates in a restricted parameter space provided that an upper bound can be given. Sometimes, in practice, it is quite difficult to provide such an upper bound. In this paper, we relax the restriction on parameter space and then try to present some useful mathematical properties of GE for the multi‐factor design problem. To obtain a GE‐based optimal design, the average prediction variance (APV) and average squared bias (ASB) are computed to form the AMSE, acting as the performance metric. The comparison between the GE and the least squares estimation (LSE) methods for a two‐factor design problem is made to validate GE's merit under the AMSE criterion. In this particular case, the comparison results show that the GE method, on the whole, outperforms the LSE method. The improvement of the GE method over the LSE method becomes more prominent if the potential bias error is significantly present. Moreover, the comparison report provided in the paper can also serve as a practical guide for response surface methodology (RSM) practitioners to choose an appropriate design point arrangement for the GE and LSE methods that achieve the objective of minimum AMSE. Copyright © 2007 John Wiley & Sons, Ltd.