Bayesian T-optimal discriminating designs

The problem of constructing Bayesian optimal discriminating designs for a class of regression models with respect to the T-optimality criterion introduced by Atkinson and Fedorov (1975a) is considered. It is demonstrated that the discretization of the integral with respect to the prior distribution leads to locally T-optimal discrimination designs can only deal with a few comparisons, but the discretization of the Bayesian prior easily yields to discrimination design problems for more than 100 competing models. A new efficient method is developed to deal with problems of this type. It combines some features of the classical exchange type algorithm with the gradient methods. Convergence is proved and it is demonstrated that the new method can find Bayesian optimal discriminating designs in situations where all currently available procedures fail.Comment: 25 pages, 3 figure

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

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-

TT-optimal designs for discrimination between two polynomial models

This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree $n-2$ and $n$. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57--70] proposed the $T$-optimality criterion for this purpose. Recently, Atkinson [MODA 9, Advances in Model-Oriented Design and Analysis (2010) 9--16] determined $T$-optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide half of the circle into equal parts if the coefficient of $x^{n-1}$ in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all $T$-optimal designs explicitly for any degree $n\in\mathbb{N}$. In particular, we show that there exists a one-dimensional class of $T$-optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of $x^{n-1}$ and $x^n$ is smaller than a certain critical value. Because of the complexity of the optimization problem, $T$-optimal designs have only been determined numerically so far, and this paper provides the first explicit solution of the $T$-optimal design problem since its introduction by Atkinson and Fedorov [Biometrika 62 (1975a) 57--70]. Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value), we propose a numerical procedure to calculate the $T$-optimal designs. The results are also illustrated in an example.Comment: Published in at http://dx.doi.org/10.1214/11-AOS956 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

TT-optimal discriminating designs for Fourier regression models

In this paper we consider the problem of constructing $T$-optimal discriminating designs for Fourier regression models. We provide explicit solutions of the optimal design problem for discriminating between two Fourier regression models, which differ by at most three trigonometric functions. In general, the $T$-optimal discriminating design depends in a complicated way on the parameters of the larger model, and for special configurations of the parameters $T$-optimal discriminating designs can be found analytically. Moreover, we also study this dependence in the remaining cases by calculating the optimal designs numerically. In particular, it is demonstrated that $D$- and $D_s$-optimal designs have rather low efficiencies with respect to the $T$-optimality criterion.Comment: Keywords and Phrases: T-optimal design; model discrimination; linear optimality criteria; Chebyshev polynomial, trigonometric models AMS subject classification: 62K0