Optimal designs for generalized linear models with biased response

This paper is concerned with the search for locally optimal designs when the observations of the response variable arise from a weighted distribution in the exponential family. Locally optimal designs are derived for regression models in which the response follows a weighted version of Normal, Gamma, Inverse Gaussian, Poisson or Binomial distributions. Some conditions are given under which the optimal designs for the weighted and original (non-weighted) distributions are the same. An efficiency study is performed to find out the behavior of the D-optimal designs for the original distribution when they are used to estimate models with weighted distributions.     

Optimal designs under two related models for interference

We consider two related models for interference: one having different directional neighbour effects and one with the same neighbour effects. We show that optimal designs for one model can be obtained from optimal designs for the other model. Acknowledgement. We are grateful to H. Kushner for advance notice of his work. SP was supported by the Isfahan University of Technology, Iran.     

Estimation and optimal designs for linear Haar-wavelet models

This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.      

Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.     

Optimal designs for a mixed interference model

This paper generalizes Kunert and Martin’s (Ann Stat 28:1728–1742, 2000) method for finding optimal designs under a fixed interference model, to find optimal designs under a mixed interference model. The results are based on the properties of information matrices in fixed and mixed models given in Markiewicz (J Stat Plan Inference 59:127–137, 1997). The method is applied to find a design which is optimal for any given variances of random neighbor effects. Research partially supported by the KBN Grant Number 5 P03A 041 21.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

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.     

Efficiencies for optimum designs when transforming the response in nonlinear models with nonconstant variance

Often the responses from mechanistic models have to be transformed to achieve error distributions that are symmetric and have constant variance. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples given of optimum designs for particular values of λ, together with the efficiency of these designs as λ varies. Approaches to finding designs robust to variations in λ are indicated and exemplified.     

Optimal money supply in models with endogenous discount factor

This paper studies the Friedman rule for the optimal quantity of money in money in the utility (MIU) and cash–credit models while considering two specifications for the endogenous discount factor. In the first specification, the discount factor depends directly on the utility level. In the second, the discount factor depends on every component of the utility function. We show that under the former specification the Friedman rule is the optimal policy. Under the latter, however, while the Friedman rule is optimal for the MIU model, it is not optimal for the cash–credit model.     

Algorithms for optimal design with application to multiple polynomial regression

The approximate theory of optimal linear regression design leads to specific convex extremum problems for numerical solution. A conceptual algorithm is stated, whose concrete versions lead us from steepest descent type algorithms to improved gradient methods, and finally to second order methods with excellent convergence behaviour. Applications are given to symmetric multiple polynomial models of degree three or less, where invariance structures are utilized. A final section is devoted to the construction of efficientexact designs of sizeN from the optimal approximate designs. For the multifactor cubic model and some of the most popular optimality criteria (D-, A-, andI-criteria) fairly efficient exact designs are obtained, even for small sample sizeN. AMS Subject Classification: 62K05.Abbreviated Title: Algorithms for Optimal Design.Invited paper presented at the International Conference on Mathematical Statistics,ProbaStat '94, Smolenice, Slovakia.     

Optimal procurement mechanisms for divisible goods with capacitated suppliers

Procurement auction literature typically assumes that the suppliers are uncapacitated [see, e.g. Dasgupta and Spulber in Inf Econ Policy 4:5–29, 1990 and Che in Rand J Econ 24(4):668–680, 1993]. Consequently, the auction mechanisms award the contract to a single supplier. We study mechanism design in a model where suppliers have limited production capacity, and both the marginal costs and the production capacities are private information. We provide a closed-form solution for the revenue maximizing direct mechanism when the distribution of the cost and production capacities satisfies a modified regularity condition [Myerson in Math Oper Res 6(1):58–73, 1981]. We also present a sealed low bid implementation of the optimal direct mechanism for the special case of identical suppliers. The results in this paper extend to other principle-agent mechanism design problems where the agents have a privately known upper bound on allocation. The authors would like to thank the anonymous referees for valuable suggestions and comments.     

Optimal Designs for Asymmetric Linear Paired Comparisons with a Profile Strength Constraint

Within the linear model framework the problem of determining optimal designs for paired comparisons of alternatives which are described by a set of discrete attributes is considered under the constraint that the alternatives in a pair are only allowed to differ with regard to a certain number of attributes. Whereas in previous treatments of this problem it was assumed that all attributes possess the same number of levels, here the general asymmetric case is discussed. We provide a characterization of optimal designs and demonstrate how this can be used to derive a solution of the design problem for many situations of interest.     

Experiments with mixtures: Optimal allocations for becker’s models

Extending Scheffé’s simplex-centroid design for experiments with mixtures, we introduce aweighted simplex-centroid design for a class of mixture models. Becker’s homogeneous functions of degree one belong to this class. By applying optimal design theory, we obtainA-, D- andI-optimal allocations of observations for Becker’s models.     

Efficient estimation and computation for the generalised additive models with unknown link function

The generalised additive models (GAM) are widely used in data analysis. In the application of the GAM, the link function involved is usually assumed to be a commonly used one without justification. Motivated by a real data example with binary response where the commonly used link function does not work, we propose a generalised additive models with unknown link function (GAMUL) for various types of data, including binary, continuous and ordinal. The proposed estimators are proved to be consistent and asymptotically normal. Semiparametric efficiency of the estimators is demonstrated in terms of their linear functionals. In addition, an iterative algorithm, where all estimators can be expressed explicitly as a linear function of $Y$, is proposed to overcome the computational hurdle for the GAM type model. Extensive simulation studies conducted in this paper show the proposed estimation procedure works very well. The proposed GAMUL are finally used to analyze a real dataset about loan repayment in China, which leads to some interesting findings.