Robust T-optimal discriminating designs

This paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the T-optimality criterion introduced by Atkinson and Fedorov [Biometrika 62 (1975) 57-70]. T-optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification. As a solution to this problem we propose a Bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the T-optimality criterion. It is shown that the corresponding Bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria. For the problem of discriminating between two polynomial regression models which differ in the degree by two the robust T-optimal discriminating designs can be found explicitly. The results are illustrated in several examples.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1117 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal experimental designs for inverse quadratic regression models

In this paper optimal experimental designs for inverse quadratic regression models are determined. We consider two different parameterizations of the model and investigate local optimal designs with respect to the $c$-, $D$- and $E$-criteria, which reflect various aspects of the precision of the maximum likelihood estimator for the parameters in inverse quadratic regression models. In particular it is demonstrated that for a sufficiently large design space geometric allocation rules are optimal with respect to many optimality criteria. Moreover, in numerous cases the designs with respect to the different criteria are supported at the same points. Finally, the efficiencies of different optimal designs with respect to various optimality criteria are studied, and the efficiency of some commonly used designs are investigated.Comment: 24 page

KL-optimum designs: theoretical properties and practical computation

In this paper some new properties and computational tools for finding KL-optimum designs are provided. KL-optimality is a general criterion useful to select the best experimental conditions to discriminate between statistical models. A KL-optimum design is obtained from a minimax optimization problem, which is defined on a infinite-dimensional space. In particular, continuity of the KL-optimality criterion is proved under mild conditions; as a consequence, the first-order algorithm converges to the set of KL-optimum designs for a large class of models. It is also shown that KL-optimum designs are invariant to any scale-position transformation. Some examples are given and discussed, together with some practical implications for numerical computation purposes.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s11222-014-9515-

\u3ci\u3eI\u3c/i\u3e-optimal or \u3ci\u3eG\u3c/i\u3e-optimal: Do We Have to Choose?

When optimizing an experimental design for good prediction performance based on an assumed second order response surface model, it is common to focus on a single optimality criterion, either G-optimality, for best worst-case prediction precision, or I-optimality, for best average prediction precision. In this article, we illustrate how using particle swarm optimization to construct a Pareto front of non-dominated designs that balance these two criteria yields some highly desirable results. In most scenarios, there are designs that simultaneously perform well for both criteria. Seeing alternative designs that vary how they balance the performance of G- and I-efficiency provides experimenters with choices that allow selection of a better match for their study objectives. We provide an extensive repository of Pareto fronts with designs for 17 common experimental scenarios for 2 (design size N = 6 to 12), 3 (N = 10 to 16) and 4 (N = 15, 17, 20) experimental factors. These, when combined with a detailed strategy for how to efficiently analyze, assess, and select between alternatives, provide the reader with the tools to select the ideal design with a tailored balance between G- and I- optimality for their own experimental situations

EE-optimal designs for second-order response surface models

$E$-optimal experimental designs for a second-order response surface model with $k\geq1$ predictors are investigated. If the design space is the $k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121-132] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $E$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $E$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem. The main difficulty in the construction of $E$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the "optimal information matrix" is larger than one (in contrast to the case $k=1$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $E$-optimal design problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1241 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Application of Advanced Stated Preference Design Methodology

This paper demonstrates the application of the design methodology developed in the Advanced Stated Preference Design project to stated preference experiments. The paper considers binary response experimental designs of two, three and four variables. In addition the special case of a two variable design with an alternative specific constant is also considered. Alternative optimality criteria are discussed. The paper concludes with recommendations on how to apply the design methodolog

Application of Advanced Stated Preference Design Methodology

This paper demonstrates the application of the design methodology developed in the Advanced Stated Preference Design project to stated preference experiments. The paper considers binary response experimental designs of two, three and four variables. In addition the special case of a two variable design with an alternative specific constant is also considered. Alternative optimality criteria are discussed. The paper concludes with recommendations on how to apply the design methodolog