© 1973 by Biometrika Trust
A new approach to mean squared error estimation of response surfaces
University of North Carolina Chapel Hill
Virginia Commonwealth University
Box & Draper (1959, 1963) and others have considered the problem of choosing a design to minimize integrated mean squared error J when the true response x'1ß1 + x'2 ß2, a polynomial of degree d2 in p variables, is approximated by a polynomial x1 b1 of lower degree d1 where b1 is the vector ß{ring}1 of usual least squares estimates. Here, the use of an estimator of the form b1 = Kß^1, where K is a diagonal matrix of appropriately chosen constants, is advocated. The particular choice of K which minimizes J will depend on the unknown elements of the parameter voctors ß and ß2. But when the parameter space can be restricted by specifying bounds, however conservative, for one or more elements of ß1 relative to 
2/N, it is possible to determine a set of K's providing smaller J than when K = I for any choice of experimental design. The special case d1 = 1 and d2 = 2 is considered in detail.
Key Words: Response surface • Multiple regression • Integrated mean squared error • Restricted parameter space • Optimal regression design