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Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes. (English) Zbl 0848.62040

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 \(1/n\).

MSC:

62K05 Optimal statistical designs
65C99 Probabilistic methods, stochastic differential equations
62Q05 Statistical tables
90C25 Convex programming
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[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
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