Abstract
We investigate the possibility of extending some results of Pázman and Pronzato (Ann Stat 42(4):1426–1451, 2014) to a larger set of optimality criteria. Namely, the problems of computing D-, A-, and \(E_k\)-optimal designs in a linear regression model are reformulated here as “infinite-dimensional” linear programming problems. The same approach is applied to combination of these optimality criteria and to the “criterion robust” problem of Harman (Metrika 60:137–153, 2004). Approximate optimum designs can then be computed by a relaxation method (Shimizu and Aiyoshi in IEEE Trans Autom Control 25(1):62–66, 1980), and this is illustrated on various examples.
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References
Atkinson AC, Donev AN (1992) Optimum experimental designs. Oxford University Press, Oxford
Atkinson AC, Chaloner K, Herzberg AM, Juritz J (1993) Optimum experimental designs for properties of a compartmental model. Biometrics 49(2):325–337
Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA (2006) Numerical optimization. Theoretical and practical aspects, 2nd edn. Springer, Heidelberg
Filová L, Harman R (2013) Criterion-robust experimental designs for the quadratic regression on a square and a cube. Commun Stat Theory Methods 42(11):2044–2055
Filová L, Trnovská M, Harman R (2012) Computing maximin efficient experimental designs using the methods of semidefinite programming. Metrika 75(5):709–719
Gaffke N (1987) Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression. Ann Stat 15(3):942–957
Gribik PR, Kortanek KO (1977) Equivalence theorems and cutting plane algorithms for a class of experimental design problems. SIAM J Appl Math 32(1):232–259
Harman R (2004) Minimal efficiency of designs under the class of orthogonally invariant information criteria. Metrika 60:137–153
Harman R, Jurík T (2008) Computing \(c\)-optimal experimental designs using the simplex method of linear programming. Comput Stat Data Anal 53:247–254
Harville DA (1997) Matrix algebra from a statistician’s perspective. Springer, New York
Heiligers B (1992) Admissible experimental designs in multiple polynomial regression. J Stat Plan Inference 31:219–233
Kelley JE (1960) The cutting-plane method for solving convex programs. J Soc Indust Appl Math 8(4):703–712
Kiefer J (1974) General equivalence theory for optimum designs (approximate theory). Ann Stat 2(5):849–879
Kiefer J (1975) Optimal design: Variation in structure and performance under change of criterion. Biometrika 62(2):277–288
Kiefer J, Wolfowitz J (1959) Optimum designs in regression problems. Ann Math Stat 30(2):271–294
Nesterov Y (2004) Introductory lectures on convex optimization: a basic course. Applied optimization, vol 87. Kluwer, Boston
Pázman A, Pronzato L (2014) Optimum design accounting for the global nonlinear behavior of the model. Ann Stat 42(4):1426–1451
Pronzato L, Pázman A (2013) Design of experiments in nonlinear models. Asymptotic normality, optimality criteria and small-sample properties. Lecture notes in statistics, vol 212. Springer, New York
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Pukelsheim F, Torsney B (1991) Optimal weights for experimental designs on linearly independent support points. Ann Stat 19(3):1614–1625
Shimizu K, Aiyoshi E (1980) Necessary conditions for min-max problems and algorithms by a relaxation procedure. IEEE Trans Autom Control 25(1):62–66
Steele JM (2004) The Cauchy–Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press, New York
Acknowledgments
We would like to thank Luc Pronzato for helpful advises. We thank also both referees for their remarks and one of them for a careful improvement of our English.
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The paper was supported by the Slovak VEGA-Grant No. 1/0521/16.
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Burclová, K., Pázman, A. Optimal design of experiments via linear programming. Stat Papers 57, 893–910 (2016). https://doi.org/10.1007/s00362-016-0782-7
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DOI: https://doi.org/10.1007/s00362-016-0782-7