Optimal designs for estimating the slope of a regression

In the common linear regression model we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest. --locally optimal design,standardized minimax optimal design,estimating derivatives,polynomial regression,Fourier regression

A note on some extremal problems for trigonometric polynomials

n.a. --Trigonometric polynomial,extremal polynomial,Chebyshev system,extremal problem

Optimal designs for estimating pairs of coefficients in Fourier regression models

In the common Fourier regression model we investigate the optimal design problem for estimating pairs of the coefficients, where the explanatory variable varies in the interval [¡¼; ¼]. L-optimal designs are considered and for many important cases L-optimal designs can be found explicitly, where the complexity of the solution depends on the degree of the trigonometric regression model and the order of the terms for which the pair of the coe±cients has to be estimated. --L-optimal designs,Fourier regression models,parameter subsets,equivalence theorem

On the Functional Approach to Optimal Designs for Nonlinear Models

This paper concerns locally optimal experimental designs for non-linear regression models. It is based on the functional approach introduced in (Melas, 1978). In this approach locally optimal design points and weights are studied as implicitly given functions of the nonlinear parameters included in the model. Representing these functions in a Taylor series enables analytical solution of the optimal design problem for many nonlinear models. A wide class of such models is here introduced. It includes, in particular,three parameters logistic distribution, hyperexponential and rational models. For these models we construct the analytical solution and use it for studying the efficiency of locally optimal designs. As a criterion of optimality the well known D-criterion is considered

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

Optimal designs for a class of nonlinear regression models

For a broad class of nonlinear regression models we investigate the local E- and c-optimal design problem. It is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the Chebyshev points, which are the local extrema of the equi-oscillating best approximation of the function f_0\equiv 0 by a normalized linear combination of the regression functions in the corresponding linearized model. The class of models includes rational, logistic and exponential models and for the rational regression models the E- and c-optimal design problem is solved explicitly in many cases.Comment: Published at http://dx.doi.org/10.1214/009053604000000382 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the functional approach to optimal designs for nonlinear models

This paper concerns locally optimal experimental designs for non- linear regression models. It is based on the functional approach intro- duced in (Melas, 1978). In this approach locally optimal design points and weights are studied as implicitly given functions of the nonlinear parameters included in the model. Representing these functions in a Taylor series enables analytical solution of the optimal design prob- lem for many nonlinear models. A wide class of such models is here introduced. It includes, in particular,three parameters logistic distri- bution, hyperexponential and rational models. For these models we construct the analytical solution and use it for studying the e_ciency of locally optimal designs. As a criterion of optimality the well known D-criterion is considered. --nonlinear regression,experimental designs,locally optimal designs,functional approach,three parameters logistic distribution,hyperexponential models,rational models,D-criterion,implicit function theorem

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

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