Some optimal designs for comparing a set of test treatments with a set of controls

In this paper, we give a detailed study of the problem of optimally comparing a set of t test treatments to a set of s controls under a 0-way elimination of heterogeneity model. The relationships between designs that are A and MV-optimal for comparing the test treatments to the controls and those that are A and MV-optimal for comparing all treatments are also studied.Research is sponsored by NSF Grant No. DMS-8700945.     

Local c- and E-optimal Designs for Exponential Regression Models

In this paper we investigate local E- and c-optimal designs for exponential regression models of the form . We establish a numerical method for the construction of efficient and local optimal designs, which is based on two results. First, we consider for fixed k the limit μ _i → γ (i = 1, ... , k) and show that the optimal designs converge weakly to the optimal designs in a heteroscedastic polynomial regression model. It is then demonstrated that in this model the optimal designs can be easily determined by standard numerical software. Secondly, it is proved that the support points and weights of the local optimal designs in the exponential regression model are analytic functions of the nonlinear parameters μ ₁, ... , μ _k . This result is used for the numerical calculation of the local E-optimal designs by means of a Taylor expansion for any vector (μ ₁, ... , μ _k ). It is also demonstrated that in the models under consideration E-optimal designs are usually more efficient for estimating individual parameters than D-optimal designs.     

Large sets of <Emphasis Type="Italic">t</Emphasis>-designs from groups

This paper addresses questions related to the existence and construction of large sets of t-(v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa’s 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of “highlight” topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them. (Communicated by Peter Horák)     

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).     

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.     

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 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.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

The expected value models on Sugeno measure space

Uncertain programming is a theoretical tool to handle optimization problems under uncertain environment. The research reported so far is mainly concerned with probability, possibility, or credibility measure spaces. Up to now, uncertain programming realized in Sugeno measure space has not been investigated. The first type of uncertain programming considered in this study and referred to as an expected value model optimizes a given expected objective function subject to some expected constraints. We start with a concept of the Sugeno measure space. We revisit some main properties of the Sugeno measure and elaborate on the g_λ random variable and its characterization. Furthermore, the laws of the large numbers are discussed based on this space. In the sequel we introduce a Sugeno expected value model (SEVM). In order to construct an approximate solution to the complex SEVM, the ideas of a Sugeno random number generation and a Sugeno simulation are presented along with a hybrid approach.     

L 1-optimization in linear systems with bounded controls

In a linear control system over a fixed time interval, we treat steering to the origin by controls (i) having realistic constraints on their values, while minimizing theL ₁-cost of control, (ii) without such constraints, and (iii) minimizing the coordinate bound, with or without a givenL ₁-cost restriction. Existence (also, nonexistence) results are obtained, together with further necessary conditions for control optimality.The author was introduced to these problems by Professors P. Hagedorn and W. Krabs and profited much by discussions with them and with other workers at the TH Darmstadt. The incisive idea that feedback controls still exist if state space is augmented by one dimension (see Example, Section 1) is due to Professor Hagedorn and obviously merits further treatment. Professor E. N. Chukwu pointed out an error in a preprint version of this paper, and the present formulation of Theorem 4.6 is the result of discussions with him. In a colloquium at Würzburg University, Professor P. C. Parks made the point that a portion ofmodern control theory is implicit in the work of Achiezer: in connection with the present material, the so-called M-moment problem, and our Theorem 4.6. The referee kindly pointed out the reference for the example in Section 1. This paper is a revised version of a report, Preprint No. 279, prepared while the author held a Humboldt Award at the TH Darmstadt.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Bounds on the number of Hadamard designs of even order

A new lower bound on the number of non‐isomorphic Hadamard symmetric designs of even order is proved. The new bound improves the bound on the number of Hadamard designs of order 2n given in [12] by a factor of 8n ? 1 for every odd n > 1, and for every even n such that 4n ? 1 > 7 is a prime. For orders 8, 10, and 12, the number of non‐isomorphic Hadamard designs is shown to be at least 22,478,260, 1.31 × 10¹⁵, and 10²⁷, respectively. For orders 2n = 14, 16, 18 and 20, a lower bound of (4n ? 1)! is proved. It is conjectured that (4n ? 1)! is a lower bound for all orders 2n ≥ 14. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 363‐378, 2001     

A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator

The Assmus–Mattson theorem is known as a method to find designs in linear codes over a finite field. It is an interesting problem to find an analog of the theorem for Z ₄-codes. In a previous paper, the author gave a candidate of the theorem. The purpose of this paper is to give an improvement of the theorem. It is known that the lifted Golay code over Z ₄ contains 5-designs on Lee compositions. The improved method can find some of those without computational difficulty and without the help of a computer.     

A process variability control chart

In this study a Shewhart type control chart namely the V _t chart, is proposed for improved monitoring of the process variability of a quality characteristic of interest Y. The proposed control chart is based on the ratio type estimator of the variance using a single auxiliary variable X. It is assumed that (Y, X) follows a bivariate normal distribution. The design structure of the V _t chart is developed for Phase-I quality control and its comparison is made with those of the S ² chart (a well-known Shewhart control chart) and the V _r chart (a Shewhart type control chart proposed by Riaz (Comput Stat, 2008a) used for the same purpose. It is observed that the proposed V _t chart outperforms the S ² and V _r charts, in terms of discriminatory power, for detecting moderate to large shifts in the process variability. It is observed that the performance of the V _t chart keeps improving with an increase in |ρ _yx | , where ρ _yx is the correlation between Y and X.     

Robust and Efficient Parametric Estimation for Censored Survival Data

We fit parametric models to survival data in the case of censoring and (outlier) contamination. To do so, we adapt the robust density power divergence methodology of Basu, Harris, Hjort, and Jones (Biometrika, 85, 549–559, 1998) to the case of censored survival data. Asymptotic properties, simulation performance and application to data are provided.     

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.     

Connections between stability and asymptotic stability of nonlinear switched systems

Here are considered nonlinear switched systems in which the switching occurs among a class of subsystems that are characterized by input–output properties stated in terms of L_p spaces of signals. The relationships between the L_p stability of each subsystem and the internal stability of the switched system are studied. In particular, conditions on the dwell time of the switching signals that guarantee the asymptotic stability of the overall system are provided. The connections among these conditions and the L_p input–output properties of the subsystems are investigated.     

Exponentially Many Hadamard Designs

If there is a Hadamard design of order n, then there are at least 2^{8n−16−9log n} non-isomorphic Hadamard designs of order 2n. Mathematics Subject Classificaion 2000: 05B05     

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.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.