Constructing optimal designs with constraints. (English) Zbl 1089.62089
Summary: We construct constrained approximate optimal designs by maximizing a criterion subject to constraints. We approach this problem by transforming the constrained optimization problem to one of maximizing three functions of the design weights simultaneously. We used a class of multiplicative algorithms, indexed by a function \(f({\cdot})\). These algorithms are shown to satisfy the basic constraints on the design weights of nonnegativity and summation to unity. We also investigate techniques for improving convergence rates by means of some suitable choices of the function \(f({\cdot})\).
MSC:
62K05 | Optimal statistical designs |
62F30 | Parametric inference under constraints |
65K10 | Numerical optimization and variational techniques |
Keywords:
Constrained optimality; Constraints; Directional derivatives; Lagrangian approach; Multiplicative algorithms; Optimality conditionsReferences:
[1] | Alahmadi, A.M., 1993. Algorithms for the construction of constrained and unconstrained optimal designs. Ph.D. Thesis, University of Glasgow, UK.; Alahmadi, A.M., 1993. Algorithms for the construction of constrained and unconstrained optimal designs. Ph.D. Thesis, University of Glasgow, UK. |
[2] | Cook, D.; Fedorov, V., Constrained optimization of experimental design, Statistics, 26, 129-178 (1995) · Zbl 0812.62080 |
[3] | Fedorov, V.V., Hackl, P., 1997. Model-Oriented Design of Experiments, Springer, New York.; Fedorov, V.V., Hackl, P., 1997. Model-Oriented Design of Experiments, Springer, New York. · Zbl 0878.62052 |
[4] | Kiefer, J., General Equivalence theory for optimum designs (approximate theory), Annals of Statistics, 2, 849-879 (1974) · Zbl 0291.62093 |
[5] | Lee, C., Contribution to discussion of ‘Constrained optimization of experimental design’ by Cook and Fedorov, Statistics, 26, 162-166 (1995) |
[6] | Mandal, S., 2000. Construction of optimising distributions with applications in estimation and optimal design. Ph.D. Thesis, University of Glasgow, UK.; Mandal, S., 2000. Construction of optimising distributions with applications in estimation and optimal design. Ph.D. Thesis, University of Glasgow, UK. |
[7] | Mandal, S., Carriere, K.C., 2001. Constructing Optimal Designs With Constraints. Technical Paper #01.04, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada.; Mandal, S., Carriere, K.C., 2001. Constructing Optimal Designs With Constraints. Technical Paper #01.04, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada. · Zbl 1089.62089 |
[8] | Mandal, S., Torsney, B., 2004. Construction of optimal designs using a clustering approach. J. Statist. Planning Inference, in revision.; Mandal, S., Torsney, B., 2004. Construction of optimal designs using a clustering approach. J. Statist. Planning Inference, in revision. · Zbl 1078.62079 |
[9] | Pukelsheim, F.; Rosenberger, J. L., Experimental designs for model discrimination, J. Amer. Statist. Assoc, 88, 642-649 (1993) · Zbl 0773.62052 |
[10] | Silvey, S. D.; Titterington, D. M.; Torsney, B., An algorithm for optimal designs on a finite design space, Comm. Statist. A, 14, 1379-1389 (1978) · Zbl 0389.62061 |
[11] | Torsney, B., Contribution to discussion of ‘maximum likelihood estimation via the EM algorithm’ by Dempster et al, J. Roy. Statist. Soc. B, 39, 26-27 (1977) |
[12] | Torsney, B., Computing optimizing distributions with applications in design, estimation and image processing, (Dodge, Y.; Fedorov, V. V.; Wynn, H. P., Optimal Design and Analysis of Experiments (1988), Elsevier Science: Elsevier Science North-Holland, Amsterdam), 361-370 |
[13] | Torsney, B., Mandal, S., 2001. Construction of constrained optimal designs. Optimum Design 2000, Kluwer Academic Publishers, Dordrecht, pp. 141-152.; Torsney, B., Mandal, S., 2001. Construction of constrained optimal designs. Optimum Design 2000, Kluwer Academic Publishers, Dordrecht, pp. 141-152. · Zbl 1423.62089 |
[14] | Whittle, P., Some general points in the theory of optimal experimental design, J. Roy. Statist. Soc. B, 35, 123-130 (1973) · Zbl 0282.62065 |
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.