Maximal determinant over a certain class of matrices and its application to D-optimality of designs
In the first part of the paper the problem of maximizing of determinants is considered. Firstly the determinants of \(t\times t\) matrices of type \(\alpha I_t-(P+P^T)\) are considered, where \(I_t\) is an identity matrix, \(P_t\) is a permutation matrix and \(\alpha>2\) is a scalar. The authors find the permutation matrix \(P\) that maximizes this determinant. Secondly the determinants of \(t\times t\) matrices of type \(\alpha I_t+P+P^T\) are considered, where \(P_t\) is a permutation matrix with zero diagonal and \(\alpha>2,5\) is a scalar. The authors also find the permutation matrix with zero diagonal that maximizes this determinant. In the second part of the paper these results are used to characterize \(D\)-optimal complete block designs under an interference model. The criterion is based on the eigenvalues of the information matrices of designs. Previously in the literature only the problem of determining \(D\)-optimal designs under a model with block effects as the only nuisance parameters over the class of designs were considered [\textit{J. A. John} and \textit{T. J. Mitchell}, ``Optimal incomplete block designs'', J. R. Stat. Soc., Ser. B 39, 39--43 (1977; Zbl 0354.05015); \textit{N. Gaffke}, ``D-optimal block designs with at most six varieties'', J. Stat. Plann. Inference 6, 183--200 (1982; Zbl 0481.62058); \textit{K. Balasubramanian} and \textit{A. Dey}, ``D-optimal designs with minimal and nearly minimal number of units'', J. Stat. Plann. Inference 52, No. 2, 255--262 (1996; Zbl 0846.62055)].
scientific article; zbMATH DE number 6013292