# Minimax designs for estimating the optimum point in a quadratic response surface

01 May 1992-Journal of Statistical Planning and Inference (North-Holland)-Vol. 31, Iss: 2, pp 235-244

TL;DR: In this paper, the authors consider designs when interest is in estimating the optimal factor combination in a multiple quadratic regression setup, supposing that this factor combination belongs to a given set.

Abstract: We consider designs when interest is in estimating the optimal factor combination in a multiple quadratic regression setup, supposing that this factor combination belongs to a given set. By involving the concepts of admissibility and invariance of designs we substantially reduce the problem of calculating minimax designs. Exemplary, we give optimal designs for some setups on the ball and on the cube.

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TL;DR: For estimating a nonlinear aspect of a linear model maximin efficient designs are derived for the case that the support of the regarded designs is given and provides linearly independent regressors as mentioned in this paper.

Abstract: For estimating a nonlinear aspect of a linear model maximin efficient designs are derived for the case that the support of the regarded designs is given and provides linearly independent regressors. Besides a general result two special results are presented, which provide maximin efficient designs under simple conditions. One of these results gives a simple condition so that the uniform design is maximin efficient. The other result deals with designs with a two-point support. These results have many applications. This is demonstrated by several examples including the problems of estimating the maximum point and the maximum value of quadratic response functions, the linear calibration problem, the problem of regression-based ratio estimation and the problems of estimating the relative effect of an additional factor and the equivalence of two treatments. Some of these examples repeat results which were already derived in the literature by straightforward methods. At last the application of the results to nonlinear models is shortly discussed.

71 citations

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01 Jan 2001

TL;DR: In this article, Torsney, N. Tachman, S.Tack, M.M. Titterington, and N.Tachman presented a list of optimal design criteria for Bayesian prediction.

Abstract: Preface. Part I: Theory. Some History Leading to Design Criteria for Bayesian Prediction A.C. Atkinson, V.V. Fedorov. Optimal Designs for the Evaluation of an Extremum Point R.C.H. Cheng, et al. On Regression Experiment Design in the Presence of Systematic Error S.M. Ermakov. Groebner Basis Methods in Mixture Experiments and Generalisations B. Giglio, et al. Efficient Designs for Paired Comparisons with a Polynomial Factor H. Grossmann, et al. On Generating and Classifying All qn-m Regular Designs for Square-Free q P.J. Laycock, P.J. Rowley. Second-Order Optimal Sequential Tests M.B. Malyutov, I.I. Tsitovich. Variational Calculus in the Space of Measures and Optimal Design I. Molchanov, S. Zuyev. On the Efficiency of Generally Balanced Designs Analysed by Restricted Maximum Likelihood H. Monod. Concentration Sets, Elfving Sets and Norms in Optimum Design A. Pazman. Sequential Construction of an Experimental Design from an I.I.D. Sequence of Experiments without Replacement L. Pronzato. Optimal Characteristic Designs for Polynomial Models J.M. Rodriguez-Diaz, J. Lopez-Fidalgo. A Note on Optimal Bounded Designs M. Sahm, R. Schwabe. Construction of Constrained Optimal Designs B. Torsney, S. Mandal. Part II: Applications. Pharmaceutical Applications of a Multi-Stage Group Testing Method B. Bond, et al. Block Designs for Comparison of Two Test Treatments with a Control S.M. Bortnick, et al. Optimal Sampling Design with Random Size Clusters for a Mixed Model with Measurement Errors A. Giovagnoli, L. Martino. Optimizing a Unimodal Response Function for Binary Variables J. Hardwick, Q.F. Stout. An Optimizing Up-And-Down Design E.E. Kpamegan, N.Flournoy. Further Results on Optimal and Efficient Designs for Constrained Mixture Experiments R.J. Martin, et al. Coffee-House Designs W.G. Muller. (D,t, C)-Optimal Run Orders L. Tack, M. Vandebroek. Optimal Design in Flexible Models, including Feed-Forward Networks and Nonparametric Regression D.M. Titterington. On Optimal Designs for High Dimensional Binary Regression Models B. Torsney, N. Gunduz. Planning Herbicide Dose-Response Bioassays Using the Bootstrap S.S. Zocchi, C.G. Borges Demetrio. Photo Gallery. Optimum Design 2000: List of Participants.

42 citations

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TL;DR: In this paper, the authors try to find optimum designs for the estimation of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex region.

Abstract: In a mixture experiment, the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe [1958. Experiments with mixtures. J. Roy. Statist. Soc. B 20, 344–360; 1963. Simplex—centroid design for experiments with mixtures. J. Roy. Statist. Soc. B 25, 235–263] first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. However, in a mixture experiment, often one is more interested in the optimum proportion of ingredients. In this paper, we try to find optimum designs for the estimation of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex region.

28 citations

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TL;DR: In this paper, the authors illustrate how certain design problems can be simplified by reparametrization of the response function, which provides further insights than the more traditional approaches, like minimax, Bayesian or sequential techniques.

Abstract: In this paper we illustrate how certain design problems can be simplified by reparametrization of the response function. This alternative viewpoint provides further insights than the more traditional approaches, like minimax, Bayesian or sequential techniques. It will also improve a practitioner’s understanding of more general situations and their “classical” treatment.

22 citations

### Cites background from "Minimax designs for estimating the ..."

...Fedorov and W. G. Miiller and Mandal and Heiligers (1992) ....

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TL;DR: In this paper, the problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model, and locally D-optimal designs are found in an explicit form, which is a generalization of the results of Fedorov and Muller (1997) for onedimensional quadratic regression function in the unit segment.

Abstract: This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Muller (1997) for onedimensional quadratic regression function in the unit segment.

21 citations

### Cites methods from "Minimax designs for estimating the ..."

...Theorem 3. For the problem in the set X and for b ¼ be1 an optimal design is given by formula ( 10 ) and its matrix M � 1 b is given by formula (11)....

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##### References

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06 Apr 2011

TL;DR: In this paper, Doubly Stochastic Matrices and Schur-Convex Functions are used to represent matrix functions in the context of matrix factorizations, compounds, direct products and M-matrices.

Abstract: Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.

6,641 citations

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TL;DR: In this article, it was shown that locally optimal designs for large numbers of experiments can be approximated by selecting a certain set of randomized experiments and by repeating each of these randomized experiments in certain specified proportions.

Abstract: It is desired to estimate $s$ parameters $\theta_1, \theta_2, \cdots, \theta_s.$ There is available a set of experiments which may be performed. The probability distribution of the data obtained from any of these experiments may depend on $\theta_1, \theta_2, \cdots, \theta_k, k \geqq s.$ One is permitted to select a design consisting of $n$ of these experiments to be performed independently. The repetition of experiments is permitted in the design. We shall show that, under mild conditions, locally optimal designs for large $n$ may be approximated by selecting a certain set of $r \leqq k + (k - 1) + \cdots + (k - s + 1)$ of the experiments available and by repeating each of these $r$ experiments in certain specified proportions. Examples are given illustrating how this result simplifies considerably the problem of obtaining optimal designs. The criterion of optimality that is employed is one that involves the use of Fisher's information matrix. For the case where it is desired to estimate one of the $k$ parameters, this criterion corresponds to minimizing the variance of the asymptotic distribution of the maximum likelihood estimate of that parameter. The result of this paper constitutes a generalization of a result of Elfving [1]. As in Elfving's paper, the results extend to the case where the cost depends on the experiment and the amount of money to be allocated on experimentation is determined instead of the sample size.

615 citations

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TL;DR: In this paper, the authors present an introduction to matrices with applications in statistics, and present a set of matrices that can be used in statistics applications in the field of computer vision.

Abstract: (1970). Introduction to Matrices with Applications in Statistics. Technometrics: Vol. 12, No. 4, pp. 929-931.

518 citations

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TL;DR: In this paper, a criterion for the choice of n experimental runs in multiresponse situations, after N runs are already available, is developed, applied to three examples involving two particular non-linear multiple response models and the results are discussed.

Abstract: : A criterion is developed for the choice of n experimental runs in multiresponse situations, after N runs are already available. The criterion is applied to three examples involving two particular non-linear multiple response models and the results are discussed.

145 citations

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23 citations