Abstract
In the paper we solve the problem of D ℋ-optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D ℋ-optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D ℋ-optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D ℋ-optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.
[1] BHATIA, R.: Matrix Analysis. Grad. Texts in Math. 169, Springer-Verlag, New York, 1996. Search in Google Scholar
[2] HARMAN, R. PRONZATO, L.: Improvements on removing nonoptimal support points in D-optimum design algorithms, Statist. Probab. Lett. 77, (2007), 90–94. http://dx.doi.org/10.1016/j.spl.2006.05.01410.1016/j.spl.2006.05.014Search in Google Scholar
[3] KIEFER, J. WOLFOWITZ, J.: The equivalence of two extremum problems, Canad. J. Math. 12, (1960), 363–366. 10.4153/CJM-1960-030-4Search in Google Scholar
[4] PÁZMAN, A.: Foundations of Optimum Experimental Design, Reidel, Dordrecht, 1986. Search in Google Scholar
[5] PUKELSHEIM, F.: Optimal Design of Experiments, John Wiley & Sons, New York, 1993. Search in Google Scholar
[6] TITTERINGTON, D.: Algorithms for computing D-optimal designs on a finite design space. In: Proceedings of the 1976 Conference on Information Science and Systems. Department of Electronic Engineering, John Hopkins University, Baltimore, 1976, pp. 213–216. Search in Google Scholar
[7] TORSNEY, B.: A moment inequality and monotonicity of an algorithm. In: Proceedings of the International Symposium on Semi-Infinite Programming and Applications (K. Kortanek, A. Fiacco, eds.), Springer-Verlag, Heidelberg, 1983, pp. 249–260. 10.1007/978-3-642-46477-5_17Search in Google Scholar
[8] ZHANG, F.: Matrix Theory, Springer-Verlag, New York, 1999. Search in Google Scholar
© 2009 Mathematical Institute, Slovak Academy of Sciences
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