Enhanced design efficiency through least upper bounds

Abstract

Lower and upper spectral bounds are known for positive-definite matrices in under Loewner (Uber monotone Matrixfunktionen. Math Z. 1934;38:177–216) ordering. Lower and upper singular bounds for matrices of order in derive under an induced ordering. These orderings are combined here to the following effects. Given two first-order experimental designs in their upper singular bound enhances both and in that its Fisher Information matrix dominates those for both and thus ordering essentials in Gauss–Markov estimation. Moreover, if and are dispersion matrices for linear estimators under and respectively, then is the spectral lower bound for in . In essence this algorithm identifies elements in complementary to those of and combines these into . Case studies illustrate gains to be made thereby in first and second-order designs. Specifically, two examples demonstrate that designs optimal under separate criteria may be combined into a single design dominating both. In addition, selected examples demonstrate that classical second-order designs may be improved inter se.