Enhanced design efficiency through least upper bounds

Abstract

Lower and upper spectral bounds are known for positive-definite $(k\times k)$ matrices in $({\mathbb{S}}_k^{+},{⪰}_L)$ under Loewner (Uber monotone Matrixfunktionen. Math Z. 1934;38:177–216) ordering. Lower and upper singular bounds for matrices of order $(n\times k)$ in $({\mathbb{F}}_{n\times k},⪰)$ derive under an induced ordering. These orderings are combined here to the following effects. Given two first-order experimental designs $(\mathit{X},\mathit{Z})$ in $({\mathbb{F}}_{n\times k},⪰),$ their upper singular bound ${\mathit{X}}_M$ enhances both $\mathit{X}$ and $\mathit{Z}$ in that its Fisher Information matrix dominates those for both $\mathit{X}$ and $\mathit{Z},$ thus ordering essentials in Gauss–Markov estimation. Moreover, if $\mathbf{\Sigma },\mathbf{\Omega },$ and $\mathbf{\Xi }$ are dispersion matrices for linear estimators under $\mathit{X},\mathit{Z},$ and ${\mathit{X}}_M,$ respectively, then $\mathbf{\Xi }$ is the spectral lower bound for $(\mathbf{\Sigma },\mathbf{\Omega })$ in $({\mathbb{S}}_k^{+},{⪰}_L)$. In essence this algorithm identifies elements in $\mathit{Z}$ complementary to those of $\mathit{X},$ and combines these into ${\mathit{X}}_M$. 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.

Keywords:

• symmetrical and rectangular matrix extremes
• monotone functions
• efficiency indices
• design augmentation

Acknowledgements

Valuable comments followed the presentation of [2] 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.