Gaffke, Norbert; Heiligers, Berthold Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes. (English) Zbl 0848.62040 J. Stat. Plann. Inference 47, No. 3, 347-376 (1995). Summary: We consider the problem of computing numerically optimal approximate designs for cubic multiple regression in \(v \geq 2\) variables under a given convex and differentiable optimality criterion, where the experimental region is either a ball or a symmetric cube, both centered at zero. We restrict to designs which are invariant with respect to the group of permutations and sign changes acting on the experimental region. For many optimality criteria this is justified by the fact that within the class of all designs there exists an optimal one which is invariant. In particular, this holds true for any convex and orthogonally invariant criterion, including Kiefer’s \(\Phi_p\)-criteria, but also for integrated variance criteria (\(I\)-criteria) and for mixtures thereof.Based on the mathematical tools developed by the authors [Metrika 42, No. 1, 29-48 (1995; Zbl 0817.62059)], we show how the algorithms of N. Gaffke and R. Mathar [Optimization 24, No. 1-2, 91-126 (1992; Zbl 0817.90075)] can be applied to compute (nearly) optimal designs. As examples, we present numerical results for (mixtures of) Kiefer’s \(\Phi_p\)-criteria under a fixed regression model, which show that the proposed methods work well.For the case that the experimental region is a ball (centered at zero) we also apply the algorithms to the subclass of (third order) rotatable designs, and evaluate the efficiency loss of optimal rotatable designs. Finally, we examine several strategies for obtaining exact designs from optimal approximate ones by changing the weights to integer multiples of \(1/n\). Cited in 2 ReviewsCited in 10 Documents MSC: 62K05 Optimal statistical designs 65C99 Probabilistic methods, stochastic differential equations 62Q05 Statistical tables 90C25 Convex programming Keywords:robust designs; multiple moments; multiple polynomial regression; invariant criterion; mixture criterion; convex optimization; line search; gradient methods; quasi-Newton methods; optimal approximate designs; cubic multiple regression; group of permutations; sign changes; rotatable designs; exact designs Citations:Zbl 0817.90075; Zbl 0817.62059 PDFBibTeX XMLCite \textit{N. Gaffke} and \textit{B. Heiligers}, J. Stat. Plann. Inference 47, No. 3, 347--376 (1995; Zbl 0848.62040) Full Text: DOI References: [1] Atwood, C. L., Optimal and efficient designs of experiments, Ann. Math. Statist., 40, 1570-1602 (1969) · Zbl 0182.51905 [2] Balinski, M. L.; Young, H. P., Fair Representation, (Meeting the Ideal of One Man, One Vote (1982), Yale University Press: Yale University Press New Haven, CT) · Zbl 0575.90004 [3] Box, G. E.P.; Draper, N. R., Empirical Model-Building and Response Surface (1987), Wiley: Wiley New York · Zbl 0482.62065 [4] Fletcher, R., Practical Methods of Optimization (1987), Wiley: Wiley New York · Zbl 0905.65002 [5] Gaffke, N.; Heiligers, B., Second order methods for solving extremum problems from optimal linear regression design, (Report No. 531, Forschungsschwerpunkt der DFG “Anwendungsbezogene Optimierung und Steuerung” (1994), Universität Augsburg) · Zbl 0863.90117 [6] Gaffke, N.; Heiligers, B., Algorithms for optimal design with application to multiple polynomial regression, Metrika 51 (1994), (To appear) · Zbl 0863.90117 [7] Gaffke, N.; Heiligers, B., Invariant approximate multiple cubic regression designs with minimal support (1994), Manuscript, Universität Augsburg [8] Gaffke, N.; Heiligers, B., Optimal and robust invariant designs for cubic multiple regression, Metrika, 42, 29-48 (1995) · Zbl 0817.62059 [9] Gaffke, N.; Mathar, R., On a class of algorithms from experimental design theory, Optimization, 24, 91-126 (1992) · Zbl 0817.90075 [10] Galil, Z.; Kiefer, J., Extrapolation designs and \(ф_p\)-optimum designs for cubic regression on the \(q\)-ball, J. Statist. Plann. Inference, 3, 27-38 (1979) · Zbl 0412.62055 [11] Hardin, R. H.; Sloane, N. J.A., A new approach to the construction of optimal designs, J. Statist. Plann. Inference, 37, 339-369 (1993) · Zbl 0799.62082 [12] Heiligers, B., Admissibility of experimental designs in linear regression with constant term, J. Statist. Plann. Inference, 28, 107-123 (1991) · Zbl 0734.62079 [13] Heiligers, B.; Schneider, K., Invariant admissible and optimal designs in cubic regression on the \(v\)-ball, J. Statist. Plann. Inference, 31, 113-125 (1992) · Zbl 0766.62044 [14] Higgins, J. E.; Polak, E., Minimizing pseudoconvex functions on convex compact sets, J. Optim. Theory App., 65, 1-27 (1990) · Zbl 0672.90093 [15] Lim, Y. B.; Studden, W. J., Efficient \(D_s\)-optimal designs for multivariate polynomial regression on the \(q\)-cube, Ann. Statist., 16, 1225-1240 (1988) · Zbl 0664.62075 [16] Pukelsheim, F., Optimal Design of Experiments (1993), Wiley: Wiley New York · Zbl 0834.62068 [17] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0229.90020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.