The Admissibility of a Response Surface Design

The usual concept of the admissibility of an experimental design (e.g., see [6], pp. 800-812) is based solely on variance considerations and no allowance is made for the possible occurrence of bias errors resulting from the use of an incorrect model. This limitation is not entirely desirable in a response surface setting where a graduating function Y(:~;Q) such as a polynomial will always fail to represent exactly a response surface n(~;§). In fact, several pairs of authors such as Box and Draper ([1], [2]) and Draper and Lawrence ([3], [4]) have demonstrated in typical situations the overriding importance of bias considerations when selecting a design to minimize expected mean-square error E(9(~;Q)-n(~;~))2 averaged over a compact set X. Motivated by their findings, this author first provides a generalization of a result on the minimization of integrated squared bias which was given in [1] and then uses it along with some ideas from decision theory to develop the notions of variance 01-), bias (B-) and mean-square (MS-) admissibility, which have particular appeal in a response surface framework. A theorem is proved characterizing a meaningful situation in which a design can be said to possess one of these types of admissibility.