Robust TTT-optimal discriminating designs (original) (raw)
Related papers
SFB 823 Robust T-optimal discriminating designs
2012
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 (1975a). T -optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification. As a solution of 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. AMS Subject Classification: 62K05
T -optimal designs for discrimination between two polynomial models
The Annals of Statistics, 2012
The 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 proposed the T -optimality criterion for this purpose. Recently Atkinson (2010) 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 a 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 ∈ 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 . 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.
The Non-Uniqueness of Some Designs for Discriminating Between Two Polynomial Models in One Variable
2010
T-optimum designs for discriminating between two nested polynomial regression models in one variable that differ in the presence or absence of the two highest order terms are studied as a function of the values of the parameters of the true model. For the value of the parameters corresponding to the absence of the next-highest order term, the optimum designs are not unique and can contain an additional support point. A numerical exploration of the non-uniqueness reveals a connection with D s-optimality for models which do contain the next highest term. Brief comments are given on the analysis of data from such designs
Bayesian TTT-optimal discriminating designs
The Annals of Statistics, 2015
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 discriminating design problems with a large number of model comparisons. Current methodology for the numerical construction of 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.
Efficient computation of Bayesian optimal discriminating designs
Journal of Computational and Graphical Statistics, 2016
An efficient algorithm for the determination of Bayesian optimal discriminating designs for competing regression models is developed, where the main focus is on models with general distributional assumptions beyond the "classical" case of normally distributed homoscedastic errors. For this purpose we consider a Bayesian version of the Kullback-Leibler (KL) optimality criterion introduced by López-Fidalgo et al. (2007). Discretizing the prior distribution leads to local KL-optimal discriminating design problems for a large number of competing models. All currently available methods either require a large computation time or fail to calculate the optimal discriminating design, because they can only deal efficiently with a few model comparisons. In this paper we develop a new algorithm for the determination of Bayesian optimal discriminating designs with respect to the Kullback-Leibler criterion. It is demonstrated that the new algorithm is able to calculate the optimal discriminating designs with reasonable accuracy and computational time in situations where all currently available procedures are either slow or fail.
Comparing robust properties of A, D, E and G-optimal designs
Computational Statistics & Data Analysis, 1994
We examine the A, D, E and G-efficiencies of using the optimal design for the polynomial regression model of degree k when the hypothesized model is of degree j and 1 Q j < k Q 8. The robustness properties of each of these optimal designs with respect to the other optimal&y criteria are also investigated. Relationships among these efficiencies are noted and practical implications of the results are discussed. In particular, our numerical results show E-optimal designs possess several properties not shared by the A, D and G-optimal designs.
T-optimal discriminating designs for Fourier regression models
Computational Statistics & Data Analysis, 2017
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.
T-optimum designs for model discrimination are notoriously difficult to find because of the computational difficulty involved in solving an optimization problem that involves two layers of optimization. Only a handful of analytical T-optimal designs are available for the simplest problems; the rest in the literature are found using specialized numerical procedures for a specific problem. We propose a potentially more systematic and general way for finding T-optimal designs using a Semi-Infinite Programming (SIP) approach. The strategy requires that we first reformulate the original minimax or maximin optimization problem into an equivalent semi-infinite program and solve it using an exchange-based method where lower and upper bounds produced by solving the outer and the inner programs, are iterated to convergence. A global Nonlinear Programming (NLP) solver is used to handle the subproblems, thus finding the optimal design and the least favorable parametric configuration that minimizes the residual sum of squares from the alternative or test models. We also use a nonlinear program to check the global optimality of the SIP-generated design and automate the construction of globally optimal designs. The algorithm is successfully used to produce results that coincide with several T-optimal designs reported in the literature for various types of model discrimination problems with normally distributed errors. However, our method is more general, merely requiring that the parameters of the model be estimated by a numerical optimization.
Model discrimination—another perspective on model-robust designs
Journal of Statistical Planning and Inference, 2007
Recent progress in model-robust designs has focused on maximiz-1 ing estimation capacities. However, for a given design, two competing models may be both estimable and yet difficult or impossible to discriminate in the model selection procedure. In this paper, we propose several criteria for gauging the capability of a design for model discrimination. The criteria are then used to evaluate a class of 18-run orthogonal designs in terms of their model-discriminating capabilities.
Optimal discrimination designs for semiparametric models
Biometrika
Much work on optimal discrimination designs assumes that the models of interest are fully specified, apart from unknown parameters. Recent work allows errors in the models to be nonnormally distributed but still requires the specification of the mean structures. Otsu (2008) proposed optimal discriminating designs for semiparametric models by generalizing the Kullback-Leibler optimality criterion proposed by López-Fidalgo et al. (2007). This paper develops a relatively simple strategy for finding an optimal discrimination design. We also formulate equivalence theorems to confirm optimality of a design and derive relations between optimal designs found here for discriminating semiparametric models and those commonly used in optimal discrimination design problems.