Optimal designs for minimax extrapolation

Speckman developed a minimax linear estimator robust against departures from an assumed model. Some concomitant questions of optimal design are investigated.     

Optimal partially balanced fractional 2 m 1+m 2 factorial designs of resolution IV

Summary This paper investigates some partially balanced fractional 2 ^m ₁+m ₂ factorial designs of resolution IV derived from partially balanced arrays, which permit estimation of the general mean, all main effects, all two-factor interactions within each set of them _k factors (k=1, 2) and some linear combinations of the two-factor interactions between the sets of them _k ones. In addition, optimal designs with respect to the generalized trace criterion defined by Shirakura (1976,Ann. Statist.,4, 723–735) are presented for each pair (m ₁,m ₂) with 2≦m ₁≦m ₂ andm ₁+m ₂≦6, and for values ofN (the number of observations) in a reasonable range. Partially supported in part by Grants 56530009 (C) and 57530010 (C).     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.     

Constrained optimal discrimination designs for Fourier regression models

In this article, the problem of constructing efficient discrimination designs in a Fourier regression model is considered. We propose designs which maximize the power of the F-test, which discriminates between the two highest order models, subject to the constraints that the tests that discriminate between lower order models have at least some given relative power. A complete solution is presented in terms of the canonical moments of the optimal designs, and for the special case of equal constraints even more specific formulae are available.     

Optimal and efficient designs for Gompertz regression models

Gompertz functions have been widely used in characterizing biological growth curves. In this paper we consider D-optimal designs for Gompertz regression models. For homoscedastic Gompertz regression models with two or three parameters, we prove that D-optimal designs are minimally supported. Considering that minimally supported designs might not be applicable in practice, alternative designs are proposed. Using the D-optimal designs as benchmark designs, these alternative designs are found to be efficient in general.     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

On exact D-optimal designs for regression models with correlated observations

Let ^* be an exact D-optimal design for a given regression model Y = X + Z . In this paper sufficient conditions are given for sesigning how the covariance matrix of Z may be changed so that not only ^* remains D-optimal but also that the best linear unbiased estimator (BLUE) of stays fixed for the design ^*, although the covariance matrix of Z * is changed. Hence under these conditions a best, according to D-optimality, BLUE of is known for the model with the changed covariance matrix. The results may also be considered as determination of exact D-optimal designs for regression models with special correlated observations where the covariance matrices are not fully known. Various examples are given, especially for regression with intercept term, polynomial regression, and straight-line regression. A real example in electrocardiography is treated shortly.     

The condition numbers for weighted Moore-Penrose inverse and weighted linear least squares problem

Condition numbers play an important role in numerical analysis. Classical normise condition numbers are used to measure the size of both input perturbations and output errors. In this paper, we study the weighted normwise relative condition numbers for the weighted Moore-Penrose inverse and the weighted linear least-squares (WLS) problems in the case of the full-column rank matrix. The bounds or formulas for the weighted condition numbers are presented. The obtained results can be viewed as extensions of the earlier works studied by others.     

Polynomial techniques for investigation of spherical designs

We investigate the structure of spherical τ-designs by applying polynomial techniques for investigation of some inner products of such designs. Our approach can be used for large variety of parameters (dimension, cardinality, strength). We obtain new upper bounds for the largest inner product, lower bounds for the smallest inner product and some other bounds. Applications are shown for proving nonexistence results either in small dimensions and in certain asymptotic process. In particular, we complete the classification of the cardinalities for which 3-designs on exist for n = 8, 13, 14 and 18. We also obtain new asymptotic lower bound on the minimum possible odd cardinality of 3-designs.      

Structure and automorphism groups of Hadamard designs

Let n be the order of a Hadamard design, and G any finite group. Then there exists many non-isomorphic Hadamard designs of order 2^{12|G| + 13} n with automorphism group isomorphic to G.This research was supported in part by the National Science Foundation.     

Optimal designs for estimating the coefficients of the lower frequencies in trigonometric regression models

In the common Fourier regression model we determine the optimal designs for estimating the coefficients corresponding to the lower frequencies. An analytical solution is provided which is found by an alternative characterization of c-optimal designs. Several examples are provided and the performance of the D-optimal design with respect to the estimation of the lower order coefficients is investigated. The results give a complete answer to an open question which was recently raised in the literature.     

Decomposing triples into cyclic designs

Motivated by constructing cyclic simple designs, we consider how to decomposing all the triples of Z_v into cyclic triple systems. Furthermore, we define a large set of cyclic triple systems to be a decomposition of triples of Z_v into indecomposable cyclic designs. Constructions of decompositions and large sets are given. Some infinite classes of decompositions and large sets are obtained. Large sets of small v with odd v<97 are also given. As an application, the results are used to construct cyclic simple triple systems.     

Two-dimensional minimax Latin hypercube designs

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the ?^∞-distance we are able to construct minimax Latin hypercube designs of n points, and to determine the minimal covering radius, for all n. For the ?¹-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of n. We conjecture that the obtained lower bound is attained, except for a few small (known) values of n. For the ?²-distance we have generated minimax solutions up to n=27 by an exhaustive search method. The latter Latin hypercube designs are included in the website www.spacefillingdesigns.nl.     

Quasi-affine symmetric designs

A symmetric design with parameters v = q ²(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P ₀, P ₁,..., P _q , P _q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2. Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India.     

Optimal designs for estimating the control values in multi-univariate regression models

This paper considers a linear regression model with a one-dimensional control variable x and an m-dimensional response vector . The components of are correlated with a known covariance matrix. Based on the assumed regression model, it is of interest to obtain a suitable estimation of the corresponding control value for a given target vector on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration on the performance of an estimator for the control value includes the difference of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of control point, say x₀, is defined to be the one which minimizes the weighted sum of squares of those standardized differences within the range of x. The objective of this study is to find a locally optimal design for estimating x₀, which minimizes the mean squared error of the estimator of x₀. It is shown that the optimality criterion is equivalent to a c-criterion under certain conditions and explicit solutions with dual response under linear and quadratic polynomial regressions are obtained.