Some third-order rotatable designs in three dimensions

Summary Some new third-order rotatable designs in three dimensions are derived from some of the available third-order rotatable designs in two dimensions. When these designs are used the results of the experiments performed according to the two-dimentional designs need not be discarded. Some of these designs may be performed sequentially in all three factors, starting with a one-dimensional design. Further, these third-order rotatable designs require a smaller number of points than most of the available three-dimensional third-order rotatable designs.     

Quaternion orthogonal designs from complex companion designs

The success of applying generalized complex orthogonal designs as space-time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space-time-polarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build quaternion orthogonal designs on complex variables for any even number of columns. A second related technique is applied to maximum rate complex orthogonal designs to generate an infinite family of quaternion orthogonal designs on complex variables such that the resulting designs have no zero entries. This second technique is also used to generate an infinite family of quaternion orthogonal designs defined over quaternion variables that display a regular redundancy. The proposed constructions are theoretically important because they provide the first known direct techniques for building infinite families of orthogonal designs over quaternion variables for any number of columns.     

Some properties of double designs in terms of lee discrepancy

Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.     

On the optimality conditions for the problem of minimizing a sum of euclidean norms

The problem of minimizing is considered. The subdifferential ofF(X) is computed for anyX and, as a consequence, the necessary and sufficient optimality conditions are stated in a more general form than the ones up to now known.     

Asymptotic results on a class of adaptive multi-treatment designs

The play-the-winner (PW) rule is an important method in clinical trials where patients can be assigned to one of the two treatments. In the PW rule, the probability of the next patient to be assigned to a particular treatment only depends on the response of the current patient. In this paper, we consider a general kind of PW rule for multi-treatment adaptive designs, in which the probability that a treatment is assigned to the next patient depends upon both the response of the previous patient and an estimated parameter, e.g., the observed success rate. Using this kind of adaptive designs, more information of previous stages are used to update the model at each stage, and more patients may be assigned to better treatments. The strong consistency and the asymptotic normality are established for the allocation proportions.     

Effort associated with a class of random optimization methods

When differentiability is not assumed random procedures can be successfully used to estimate the extreme values of a given function. For a class of such algorithms we treat the problem of estimating the mean effort.Research partially supported by CNPq-Brazil.     

A note on partially balanced ternary designs

Two new methods of constructing a series of partially balanced ternary designs are presented. One from a BIB design and a PBIB design, and the second from a PBIB design alone, obtained by method of differences in both the cases.     

Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

On the construction of a class of invariant polynomials in several matrices,extending the zonal polynomials

Summary The construction of a class of invariant polynomials in several matrices extending the zonal polynomials is discussed. The method adopted generalized the orginal group-theoretic approach of James [9]. A table of three-matrix polynomials up to degree 5 is presented. CSIRO     

Some bounds for partially balanced block designs

Bounds on eigenvalues of theC-matrix for a partially balanced block (PBB) design are given together with some bounds on the number of blocks. Furthermore, a certain equiblock-sized PBB design is characterized. These results contain, as special cases, the known results for variance-balanced block designs and so on.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Variance functions form-grouped cylindrically rotatable designs of type 3

Summary Formulae for variance of difference between two estimated responses are derived form-grouped first-order and 2-grouped second- and thirdorder cylindrically rotatable designs of type 3.     

On the optimality of block designs

Summary A family of highly efficient designs in the sence of theE-criterion is herein described. These designs have strictly betterE-performance than regular graph designs, yet the off-diagonal entries of theirC-matrix differ by as much as two. Some counterexamples to conjectures in experimental design are then supplied. Asymptotic behavior and equivalence of theA- andD-criteria under a certain condition of uniqueness are analyzed as well.     

Robustness of A-optimal designs

Suppose that Y=(Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j),X=(X_ij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Y_i are dependent, i.e., for i≠i′, the covariance of Y_i,Y_i^′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(X_ij), where X_ij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight b_j being weighed on the left and right of a chemical balance respectively.     

Optimal allocation of units in experimental designs with hierarchical and cross classification

Consider the class of random linear models induced by possible allocations of units in an experimental design with hierarchical or cross classification. Assuming a balanced model belongs to the class, it is shown that this model is optimal for estimation of mean.     

New insights into best linear unbiased estimation and the optimality of least-squares

New results in matrix algebra applied to the fundamental bordered matrix of linear estimation theory pave the way towards obtaining additional and informative closed-form expressions for the best linear unbiased estimator (BLUE). The results prove significant in several respects. Indeed, more light is shed on the BLUE structure and on the working of the OLS estimator under nonsphericalness in (possibly) singular models.     

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.     

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.