Exact optimality of augmented order-of-addition designs
Abstract
We prove that, for the pairwise-ordering model with at least three components, adding any run to an order-of-addition orthogonal array gives a D-optimal design among all exact designs of the enlarged size, allowing replication. We also establish the corresponding result for foldover designs: adding any reversal pair to a foldover order-of-addition orthogonal array gives a D-optimal design within the foldover class at the enlarged size. Both augmentations also minimise the average prediction variance and a covariance-scaled information discrepancy, and maximise the minimum relative efficiency. The key step represents the scaled discrepancy through signed comparisons of triples and derives a sharp integer lower bound; classical spectral inequalities then give the exact-size optima, and signed four-component relative-order counts characterise equality. Complete reversal pairs additionally protect the estimation of pairwise-ordering effects against arbitrary reversal-invariant nuisance means. At the smallest estimable five-component foldover size, exact enumeration identifies five determinant-optimal classes with different prediction and balance properties, and a seven-component example shows that deleting a reversal pair need not be optimal even when it minimises the discrepancy. Constructions through ten components illustrate the attained run sizes.
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BibTeX
@misc{georgiou2026exact,
title = {{Exact optimality of augmented order-of-addition designs}},
author = {Stelios D. Georgiou and Haralambos Evangelaras and Stella Stylianou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08793},
eprint = {2611.08793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08793},
}
RIS
TY - UNPB
TI - Exact optimality of augmented order-of-addition designs
AU - Stelios D. Georgiou
AU - Haralambos Evangelaras
AU - Stella Stylianou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08793
UR - https://arxiv.org/abs/2611.08793
AB - We prove that, for the pairwise-ordering model with at least three components, adding any run to an order-of-addition orthogonal array gives a D-optimal design among all exact designs of the enlarged size, allowing replication. We also establish the corresponding result for foldover designs: adding any reversal pair to a foldover order-of-addition orthogonal array gives a D-optimal design within the foldover class at the enlarged size. Both augmentations also minimise the average prediction variance and a covariance-scaled information discrepancy, and maximise the minimum relative efficiency. The key step represents the scaled discrepancy through signed comparisons of triples and derives a sharp integer lower bound; classical spectral inequalities then give the exact-size optima, and signed four-component relative-order counts characterise equality. Complete reversal pairs additionally protect the estimation of pairwise-ordering effects against arbitrary reversal-invariant nuisance means. At the smallest estimable five-component foldover size, exact enumeration identifies five determinant-optimal classes with different prediction and balance properties, and a seven-component example shows that deleting a reversal pair need not be optimal even when it minimises the discrepancy. Constructions through ten components illustrate the attained run sizes.
ER -