On the usage of refined linear models for determiningN-way classification designs which are optimal for comparing test treatments with a standard treatment

Summary In this paper we consider experimental settings in whichv test treatments are to be compared to some control or standard treatment and where heterogeneity needs to be eliminated inn-directions. Using techniques similar to those used by Kunnert (1983,Ann. Statist.,11, 247–257) concerning the determination of optimal designs under a refined linear model, some methods are given for constructingn-way classification designs which areA- andMV-optimal for estimating elementary treatment differences involving the standard treatment fromm-way classification designs,m<n, which areA- andMV-optimal for estimating the same treatment differences. Examples are given for the casen=2 to show how the results obtained can be applied. This research was supported by NSF grant No. DMS-8401943.     

Efficient crossover designs for comparing test treatments with a control treatment

Within a large family of crossover designs this paper characterizes the mathematical structures of A-optimal and A-efficient crossover designs for the purpose of statistical comparison between t experimental treatments with a control (standard) treatment. It further guides the user how to go about the construction of these designs and if needed doing the last minute modifications. To demonstrate the ideas some very interesting optimal and efficient small designs are constructed. The mathematical and statistical tools developed here could be very useful in other areas of design of experiments. Many interesting and not yet solved design problems for further research are implicitly stated throughout the paper.     

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.     

On the robustness of balanced fractional 2m factorial designs of resolution 2l+1 in the presence of outliers

Summary By use of the algebraic structure, we obtain a simplified expression for the outlier-insensitivity factor for balanced fractional 2^m factorial (2^m-BFF) designs of resolution 2l+1 derived from simple arrays (S-arrays), whose measure has been introduced by Ghosh and Kipnegeno (1985,J. Statist. Plann. Inference,11, 119–129). It is defined by use of the measure suggested by Box and Draper (1975,Biometrika, 62 (2), 347–352). As examples, we study the sensitivity ofA-optimal 2^m-BFF designs of resolution VII (i.e.,l=3) given by Shirakura (1976,Ann. Statist.,4, 515–531; 1977,Hiroshima Math. J.,7, 217–285). We observe that these designs are robust in the sense that they have low sensitivities. Research supported in part by Grant 59530012 (C) and 60530014 (C), Japan.     

Efficiency Bounds for Product Designs in Linear Models

We provide lower efficiency bounds for the best product design for an additive multifactor linear model. The A-optimality criterion is used to demonstrate that out bounds are better than the conventional bounds. Applications to other criteria, such as IMSE (integrated mean squared error) criterion are also indicated. In all the cases, the best product design appears to perform better when there are more levels in each factor but decreases when more factors are included. Explicit efficiency formulas for non-additive models are also constructed.     

Some constructions of PBIB designs

Summary Constructions of three series of regular GD and semi-regular GD designs are given. Furthermore, a series of rectangular PBIB designs is constructed and particular cases of this series which reduce to PBIB designs with two associate classes are also provided. Written while visiting Department of Statistics, University of Indore, India, December 1983 through February 1984.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

Some constructions of two-associate class PBIB designs

Summary Two-associate class PBIB designs, having association schemes of GD orL ₂ types, are constructed by using patterned matrices and by methods of taking unions of sets of blocks.     

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.     

Direct solution of an optimal layout problem for isotropic heat conductors with a volume fraction constraint

This paper considers the problem of the optimal layout of two isotropic heat conductors on a plane subject to a constraint on the overall volume fraction of the better conductor. The problem is solved initially using the traditionalG _m -closure technique. Later, the utility of the direct relaxation technique introduced in Ref. 1 is illustrated as we apply it to verify the original results.This paper is dedicated with gratitude by the author to his advisor K. A. Lurie. The author wishes to acknowledge discussions with D. Clomenil.     

On the Characterisation of AG(n,q) by its Parameters asa Nearly Triply Regular Design

We show that a non-symmetric nearly triply regular designD with and in which every line has at least q points is AG(n,q) for prime power q > 2 and positiveinteger n 3.     

Some Triple Block Intersection Numbers of Paley 2-Designs of QN-type

We determine here some possible values for thecardinality of the intersection of three blocks from Paley2-(2q+1, q, (q-1)/2) designs where qis a prime power such that (mod 4).     

Some Solutions for a Class of Singular Equations

In this paper we obtain all solutions which depend only on r for a class of partial differential equations of higher order with singular coefficients.     

I-convergence theorems for a class of k-positive linear operators

In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.      

Abelian and Tauberian theorems for a new trigonometric method of summation

We first introduce a new trigonometric method of summation and then prove some Abelian and Tauberian theorems for this method.     

A Generalization of a Theorem of Dehon for Simple t-Designs

A generalization of a theorem of Dehon for simple t-designs is proved. Several new families of 4- and 5-designs having certain additional property regarding their block intersection numbers are obtained by applying this result.The author acknowledges the support of SFB Forschungsschwerpunkt-Geometrie Grant, University of Heidelberg.     

Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers

We study a multi-server M/M/c type queue with a single vacation policy for some idle servers. In this queueing system, if at a service completion instant, any d (d c) servers become idle, these d servers will take one and only one vacation together. During the vacation of d servers, the other c–d servers do not take vacation even if they are idle. Using a quasi-birth-and-death process and the matrix analytic method, we obtain the stationary distribution of the system. Conditional stochastic decomposition properties have been established for the waiting time and the queue length given that all servers are busy.     

On optimal condition numbers for Markov chains

Let T and be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)–norm pair if . Here π and are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient , where e is the n-vector of all 1’s and A ^# is the group generalized inverse of A = I − T, is the smallest condition number of Markov chains with respect to the (p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm pairs. These are, respectively, κ ₃ and κ ₆ in the Cho–Meyer list of 8 condition numbers. Kirkland has studied κ ₃(T). He has shown that and he has characterized transition matrices for which equality holds. We prove here again that 2κ ₃(T) ≤ κ(6) which appears in the Cho–Meyer paper and we characterize the transition matrices T for which . There is actually only one such matrix: T = (J _n  − I)/(n − 1), where J _n is the n × n matrix of all 1’s. This research was supported in part by NSERC under Grant OGP0138251 and NSA Grant No. 06G–232.     

Optimum experimental design for a regression on a hypercube-generalization of Hoel's result

In the note Hoel's result (1965, Ann. Math. Statist., 36, 1097–1106) is generalized to a large family of experimental design optimality criterions. Sufficient conditions for optimality criterion are given, which ensure existence of the optimum experimental design measure which is a product of design measures on lower dimensional domains.     

A pathwise solution for nonlinear parabolic equations with stochastic perturbations

We analyse here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient ∂ _xu with respect to the state variable, ∈ ℝⁿ. A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite dimensional Lie algebra generated by the given diffusion vector fields.