$${\phi_p}$$-optimal designs for a linear log contrast model for experiments with mixtures

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S ^k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these -optimal designs behave. Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001.     

On Minimally-supported <Emphasis Type="Italic">D</Emphasis>-optimal Designs for Polynomial Regression with Log-concave Weight Function

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.     

A note on the uniform distribution on the arcsin points

In his book Pukelsheim [8] pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains theD-optimal arcsin support design and theD ₁-optimal design for a polynomial regression. The results allow explicit representations ofD-, andD ₁-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.     

Optimal variance- and efficiency-balanced designs for one- and two-way elimination of heterogeneity

Summary In this paper, a series ofE-optimal non-binary variance balanced (block or row-column) designs and a series ofE-optimal non-binary efficiency balanced (block or row-column) designs are provided in certain broad classes of competing designs. Furthermore, their high efficiencies by the usualA- andD-optimality criteria are shown.     

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

E-optimal minimally connected block designs under mixed effects model

Summary Considering a mixed effects model in a minimally connected block design set-up, we obtain designs which areE-optimal, uniformly in the ratio of the variance components, for inference on varietal contrasts which constitute the fixed effects in the model. Work supported by the Centre for Management and Development Studies, Indian Institute of Management Calcutta. Work supported by the National Sciences and Engineering Research Council of Canada under grant number 7272 and partially done while visiting Indian Statistical Institute, Calcutta, India.     

Elfving’s method for <Emphasis Type="Italic">m</Emphasis>-dimensional models

Elfvings method is a well known graphical procedure for obtaining c-optimal designs. Although the method is valid for any dimension it is rarely used for more than two parameters. Its usefulness seems to be constrained by the difficulty on the construction of the Elfving set. In this paper a computational procedure for finding c-optimal designs using Elfvings method for more than two dimensions is provided. It is suitable for any model, achieving an efficient performance for three or four dimensional models. The procedure can be implemented in most programming languages.Acknowledgments. The authors would like to thank Dr. Torsney for his helpful comments. This research was supported by a grant from JCyL SA004/01.     

One-step ahead adaptive D-optimal design on a finite design space is asymptotically optimal

We study the consistency of parameter estimators in adaptive designs generated by a one-step ahead D-optimal algorithm. We show that when the design space is finite, under mild conditions the least-squares estimator in a nonlinear regression model is strongly consistent and the information matrix evaluated at the current estimated value of the parameters strongly converges to the D-optimal matrix for the unknown true value of the parameters. A similar property is shown to hold for maximum-likelihood estimation in Bernoulli trials (dose–response experiments). Some examples are presented.     

sk−p Fractional factorial designs in sb blocks

Draper and Guttman (1997) shows that for basic 2^k−p designs, p≥0, k − p replicates of blocks designs of size two are needed to estimate all the usual (estimable) effects. In this work, we provide an algebraic formal proof for the two-level blocks designs results and present results applicable to the general case; that is, for the case of s ^k factorial (p=0) or s ^k−p fractional factorial (p >0) designs in s ^b blocks, where 0<b<k− p, at least replicates are needed to clear up all possible effects. Through the theoretical development presented in this work, it can provide a clearer view on why those results would hold. We will also discuss the estimation equations given in Draper and Guttman (1997).  Research supported in part by the National Science Council of Taiwan, R.O.C., Grant No. NSC 89-2118-M110-010. Acknowledgement. The authors would like to thank the referee for very helpful comments.     

Designs for estimating an extremal point of quadratic regression models in a hyperball

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 Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs

Recently, Xu and Wu (2001) presented generalized minimum aberration criterion for comparing and selecting general fractional factorial designs. This criterion is defined using a set of _u(D) values, called J-characteristics by us. In this paper, we find a set of linear equations that relate the set of design points to that of J-characteristics, which implies that a factorial design is uniquely determined by its J-characteristics once the orthonormal contrasts are designated. Thereto, a projection justification of generalized minimum aberration is established. All of these conclusions generalize the results for two-level symmetrical factorial designs in Tang (2001).Acknowledgements The authors are grateful to the editor, the associate editor and the referees for their valuable comments. This paper is supported by NNSF of P.R.China grant No. 10171051. and RFDP grant No. 1999005512.     

A general construction of E(s 2)-optimal large supersaturated designs

A general method for construction of E(s ²)-optimal, two-level supersaturated designs (SSDs) with the equal occurrence property, from supplementary difference sets is introduced. It is proved that SSDs constructed in this way are E(s ²)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of E(s ²)-optimal large SSDs.     

Exact D-optimal designs for first-order trigonometric regression models on a partial circle

Recently, various approximate design problems for low-degree trigonometric regression models on a partial circle have been solved. In this paper we consider approximate and exact optimal design problems for first-order trigonometric regression models without intercept on a partial circle. We investigate the intricate geometry of the non-convex exact trigonometric moment set and provide characterizations of its boundary. Building on these results we obtain a solution of the exact $D$ -optimal design problem. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of observations.     

A statistical treatment of the problem of division

The problem of division is one of the most important problems in the emergence of probability. It has been long considered solved from a probabilistic viewpoint. However, we do not find the solution satisfactory. In this study, the problem is recasted as a statistical problem. The outcomes of matches of the game are considered as an infinitely exchangeable random sequence and predictors/estimators are constructed in light of de Finetti representation theorem. Bounds of the estimators are derived over wide classes of priors (mixing distributions). We find that, although conservative, the classical solutions are justifiable by our analysis while the plug-in estimates are too optimistic for the winning player.Acknowledgement. The authors would like to thank the referees for the insightful and informative suggestions and, particularly, for referring us to important references.Supported by NSC-88-2118-M-259-009.Supported in part by NSC 89-2118-M-259-012.Received August 2002     

<Emphasis Type="Italic">D</Emphasis>-optimal designs for polynomial regression with exponential weight function

Weighted polynomial regression with exponential weight function on an interval is considered. The D-optimal designs are completely characterized via three differential equations. Some invariant properties of the optimal designs under affine transformation are derived. The optimal design as degree of polynomial goes to infinity, is shown to converge weakly to the arcsin distribution. Comparisons of the optimal designs with the arcsin distribution are also made.     

Manipulation via merging and splitting in claims problems

In claims problems, we study coalitional manipulations via claims merging and splitting. We characterize (division) rules that are non-manipulable via (pairwise) splitting and that also satisfy standard axioms of equal treatment of equals, consistency, and continuity. And we obtain a similar result for non-manipulability via (pairwise) merging.Received: 22 January 2002, Accepted: 5 February 2003, JEL Classification: C70, C71, D70I am grateful to William Thomson for many helpful comments and suggestions. I also thank Murat Sertel and an anonymous referee for many helpful suggestions for the revision. All remaining errors are mine.     

On the E-optimality of truncated block designs

Sufficient conditions are found for designs, derived from completely symmetric designs by deleting binary blocks, to be E-optimal. Sufficient conditions are also found for E-optimality of designs obtained from other E-optimal designs by deleting all blocks forming a balanced incomplete block design on a subset of the treatments. The results include many binary and non-binary designs for which E-optimality was previously unknown.     

Minimal efficiency of designs under the class of orthogonally invariant information criteria

Consider the linear regression model with uncorrelated errors and an experimental design . In the article, we address the problem of calculating the minimal efficiency of with respect to the class of orthogonally invariant information criteria, containing all Kiefers criteria of _p-optimality, among others. We show that the -minimal efficiency of is equal to the minimal efficiency of with respect to a finite class of criteria which generalize the criterion of E-optimality. We also formulate conditions under which a design is maximin efficient, i.e. the most efficiency-stable for criteria from . To illustrate the results, we calculated the -minimal efficiency of _p (in particular D, A and E) optimal designs for polynomial regression on [–1,1] up to degree 4. Moreover, for the quadratic model we explicitly constructed the -maximin efficient design.Acknowledgement. The author would like to thank prof. Pázman as well as an anonymous referee for useful and inspiring comments on earlier versions of this article.Supported by: VEGA grant of the Slovak Agency No. 1/0264/03     

$$\mathcal{D}$$-optimal Minimum Support Mixture Designs in Blocks

An easy method to construct efficient blocked mixture experiments in the presence of fixed and/or random blocks is presented. The method can be used when qualitative variables are involved in a mixture experiment as well. The resulting designs are -optimal in the class of minimum support designs. It is illustrated that the minimum support designs are more efficient than orthogonally blocked mixture experiments presented in the literature and only slightly less efficient than -optimal designs.     

Asymptotic distributions of smoothed histograms

As an estimator for an unknown probability density functionf, concentrated on a known intervalI, one can use a histogram smoothed by a suitable family of lattice distributions. For such an estimator a uniform weak consistency result and a central limit theorem with an error bound are given. Further for the global deviation of fromf the asymptotic distribution is developed.Partially supported by the Natural Sciences and Engineering Research Council of Canada, grant A 2983, A4806, and A3988.