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.
Acknowledgements
Valuable comments followed the presentation of [Citation2] to the Joint AMS-IMS-SIAM Summer Research Conference on Stochastic Inequalities, Seattle, Washington, July 1991. The late Professor Samuel Kotz remarked as most important the approximately three pages that served to set the stage for the present manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.