Maximin power designs in testing lack of fit

In a previous article (Wiens, 1991) we established a maximin property, with respect to the power of the test for Lack of Fit, of the absolutely continuous uniform 'design' on a design space which is a subset of R-q with positive Lebesgue measure. Here we discuss some issues and controversies surrounding this result. We find designs which maximize the minimum power, over a broad class of alternatives, in discrete design spaces of cardinality N. We show that these designs are supported on the entire design space. They are in general not uniform for fixed N, but are asymptotically uniform as N -> infinity. Several examples with N fixed are discussed; in these we find that the approach to uniformity is very quick. (C) 2018 Elsevier B.V. All rights reserved.