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Construction method for some A- and D-optimal weighing designs when N\(\equiv 3\,\,(mod\,4)\). (English) Zbl 0731.62131

Summary: Let n and N be positive integers and let \({\mathcal X}(N,n)\) be the set of all \(N\times n\) matrices X whose elements are \(+1\) or -1. If X minimizes \(tr(X'X)^{-1}(\det (X'X)^{-1})\) then X is said to be A-optimal (D- optimal) in \({\mathcal X}(N,n)\). For \(N\equiv 3\) (mod 4), a lower bound for \(tr(X'X)^{-1}(\det (X'X)^{-1})\) has been obtained by the authors [Ann. Stat. 17, No.4, 1906-1915 (1989; Zbl 0695.62190)] and it is shown that this bound is attained if X is such that \(X'X\) is a block matrix with specified blocks of only one size or of two contiguous sizes.
In this paper some A- and D-optimal X attaining the lower bound for \(n\leq N-3\) are constructed by deleting s-4 specified rows of a Hadamard matrix of order N-3 and then augmenting it by s-1 specified rows for \(5\leq s\leq 8\). Tables of A- and D-optimal designs thus constructed are given for \(N<100\).

MSC:

62K05 Optimal statistical designs
62K15 Factorial statistical designs
62K10 Statistical block designs
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)

Citations:

Zbl 0695.62190
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References:

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[7] Sathe, Y. S.; Shenoy, R. G., Optimal values of \(s\) for \(N\)≡3 (mod 4) and \(N<100\) for A-optimal weighing designs (1988), Unpublished manuscript · Zbl 0731.62131
[8] Sathe, Y. S.; Shenoy, R. G., A-optimal weighing designs when \(N\)≡3 (mod 4), Ann. Statist., 17, 4 (1989) · Zbl 0695.62190
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