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The theory of uniform design has received increasing interest because of its wide application in the field of computer experiments. The generalized discrete discrepancy is proposed to evaluate the uniformity of the mixed-level factorial design. In this paper, the authors give a lower bound of the generalized discrete discrepancy and provide some construction methods of optimal mixed-level uniform designs which can achieve this lower bound. These methods are all deterministic construction methods which can avoid the complexity of stochastic algorithms. Both saturated mixed-level uniform designs and supersaturated mixed-level uniform designs can be obtained with these methods. Moreover, the resulting designs are also Χ2-optimal and minimum moment aberration designs.

This paper investigates the optimal design problem for the prediction of the individual parameters in hierarchical linear models with heteroscedastic errors. An equivalence theorem is established to characterize D-optimality of designs for the prediction based on the mean squared error matrix. The admissibility of designs is also considered and a sufficient condition to simplify the design problem is obtained. The results obtained are illustrated in terms of a simple linear model with random slope and heteroscedastic errors.

ABSTRACTThe exponentially weighted moving average (EWMA) control chart is a very popular memory-type chart and also effective in detecting small shifts in the process mean. Several modifications of the EWMA chart, such as the double and triple EWMA charts (regarded as DEWMA and TEWMA charts, respectively) have been developed to enhance its performance in detecting small shifts. In the present article, we propose the quadruple EWMA chart (regarded as QEWMA chart) in order to improve much more the detection ability of the EWMA chart. The run-length characteristics of the proposed chart are evaluated by performing Monte Carlo simulations. Comparing with the EWMA, DEWMA and TEWMA charts, it is found that the QEWMA chart outperforms its competitors for small shifts. Moreover, it is shown that the proposed chart is more in-control (IC) robust under several non-normal distributions than the other charts, especially for a medium value of the smoothing parameter. The effect of inertia on the performance of the QEWMA chart is also investigated as a part of this article. Finally, two examples are provided to demonstrate the application of the proposed chart.

ABSTRACTQuality control charts are extensively used to monitor processes. The EWMA chart is a good alternative to the Shewhart chart to quickly detect small and moderate shifts in the process mean. The DEWMA charting procedure is an enhanced approach for the EWMA chart and performs much better, especially for small and moderate shifts. In this article, we propose the triple EWMA (TEWMA) chart in an effort to improve much more the detection ability of the classical EWMA chart. Monte Carlo simulations are used to evaluate the run-length characteristics of the proposed chart. A comparison study versus the DEWMA, EWMA and GWMA charts indicates that the TEWMA chart with time-varying control limits is more effective in detecting small shifts while it is comparable with the other charts in moderate and large shifts. Moreover, the proposed chart has better inertia properties than the competing charts and is also shown to be in-control robust for small values of the smoothing parameter. Finally, two examples are given to display the application of the TEWMA chart.

The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two-and three-level design. A formulation of wrap-around L2-discrepancy for the extended designs is developed. As a benchmark of obtaining (nearly) uniform asymmetrical extended designs, a lower bound to the wrap-around L2-discrepancy for our proposed designs is established. Thorough numerical results are displayed, which provide further corroboration to the derived theoretical results.

ABSTRACTIn recent years, both robust parameter designs (RPDs) and supersaturated designs (SSDs) have attracted a great deal of attention. In the present article, a combination of the above two techniques is considered. More precisely, we propose a construction of an effective SSD along with an analysis method in order to deal with the significant problem of the robust parameter design methodology (RPDM). Combining iterative Sure Independence Screening (SIS) variable selection and a penalized method, namely smoothly clipped absolute deviation (SCAD), we perform an analysis of the SSDs developed in the present work. The proposed methodology is applied in different models so as to show its effectiveness in many different scenarios, assuming both first-and second-order models of a response surface design. Two illustrative examples as well as numerous numerical experiments are conducted for plenty of cases. The results imply that the proposed method is highly effective for identifying the active effects of main factors, two-factor interactions, three-factor interactions, and the pure quadratic ones, under the assumption of effect sparsity.

This paper considers the derivation of a new lower bound for the generalized discrete discrepancy of designs involving mixed level factors. Using mirror mapping and foldover techniques, a simple and efficient construction method of Sudoku-based mixed level uniform designs is proposed. Moreover, the properties of such Sudoku-based mixed level uniform designs are investigated. [ABSTRACT FROM AUTHOR]

The objective of this paper is to discuss the issue of the projection uniformity of asymmetric fractional factorials. On the basis of Lee discrepancy, the authors define the projection Lee discrepancy to measure the uniformity for low-dimensional projection designs. Moreover, the concepts of uniformity pattern and minimum projection uniformity criterion are proposed, which can be used to assess and compare two and three mixed levels factorials. Statistical justification of uniformity pattern is also investigated.

This paper considers the problem of estimation of Rp= Pr(Yp> max(Y1, …, Yp-1)) where Y1; Y2; …, Ypare all independent random variables following one parameter exponential distributions. As an application, Rp is the stress-strength reliability of a component having strength Ypand is subjected to several independent stresses Y1; Y2; …, Yp-1. Apart from providing several estimators of Rp, the main thrust is on the derivation of uniformly minimum variance unbiased estimator (UMVUE). A simulation study is performed to compare the estimators obtained in this paper.

Search designs form an indispensable tool under model uncertainty. However, the progress in this area has been limited due to com- plexities of checking the necessary rank condition for search designs. Designs for searching two two-factor and one three-factor interaction effects under a tree structure in the presence of the general mean and all the main effects are provided here. These designs are obtained under the assumption that four-factor and higher order interaction effects are all absent. Probability of correct searching obtained through simulations is also discussed.

Abstract: We study the uniformity of two-level -type designs based on the centered and wrap-around -discrepancies. Based on the known formulation of the measures of uniformity, we present some new lower bounds to centered and wrap-around -discrepancies, which can be used as benchmarks in searching uniform -type designs or helping to proof that a good design is in fact uniform. Using the efficient algorithm proposed in , some two-level uniform designs are obtained. [Copyright &y& Elsevier]

Abstract: Lee discrepancy has been employed to measure the uniformity of fractional factorials. In this paper, we further study the statistical justification of Lee discrepancy on asymmetrical factorials. We will give an expression of the Lee discrepancy of asymmetrical factorials with two- and three-levels in terms of quadric form, present a connection between Lee discrepancy, orthogonality and minimum moment aberration, and obtain a lower bound of Lee discrepancy of asymmetrical factorials with two- and three-levels.

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only a few experiments during the preliminary stages of experimentation. A popular measure to assess multilevel SSDs is the E(χ2) criterion. The literature reports on SSDs have concentrated mainly on balanced designs. For s-level SSDs, the restriction of the number of runs Nbeing only a multiple of sis really not required for the purpose of use of such designs. Just like when Nis a multiple of sand the design ensures orthogonality of the factor effects with the mean effect, in the case of Nnot a multiple of s, we ensure near orthogonality of each of the factors with the mean. In this article we consider s-level E(χ2)-optimal designs for , 0. We give an explicit lower bound on E(χ2). We give the structures of design matrices that attain the lower bounds. Some combinatorial methods for constructing E(χ2)-optimal SSDs are provided.

Supersaturated designs (SSDs) offer apotentially useful way to investigate many factors with only a few experiments during the preliminary stages of experimentation. A popular measure to assess multilevel SSDs is the E(χ2) criterion. The literature reports on SSDs have concentrated mainly on balanced designs. For s-level SSDs, the restriction of the number of runs Nbeing only a multiple of sis really not required for the purpose of use of such designs. Just like when Nis a multiple of sand the design ensures orthogonality of the factor effects with the mean effect, in the case of Nnot a multiple of s, we ensure near orthogonality of each of the factors with the mean. In this article we consider s-level E(χ2)-optimal designs for N≡ n(mod s), 0≤ n≤ s− 1. We give an explicit lower bound on E(χ2). We give the structures of design matrices that attain the lower bounds. Some combinatorial methods for constructing E(χ2)-optimal SSDs are provided.

Abstract: The objective of this paper is to study the issue of discrete discrepancy [Hickernell, F.J., Liu, M.Q., 2002. Uniform designs limit aliasing. Biometrika 89, 893–904; Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003. Optimal mixed-level supersaturated design. Metrika 58, 279–291; Qin, H., Fang, K.T., 2004. Discrete discrepancy in factorial designs. Metrika 60, 59–72], which has wide application to the field of fractional factorial designs. Here we present an improved lower bound to discrete discrepancy. [Copyright &y& Elsevier]

The role of uniformity measured by the centered L2-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L2-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L2-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.The role of uniformity measured by the centered L2-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L2-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L2-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.

Rosenbaum (1996) and Hedayat and Stufken (1999) considered the construction of compound orthogonal arrays for the two-level factors which are useful in the study of dispersion experiments, pioneered by Taguchi (1986). This paper provides a general method of constructing such arrays and their applications to obtain several asymmetric compound orthogonal arrays for mixed level dispersion experiments. Such arrays are also found to be universally optimal even if the model includes some more parameters.

Optimality of orthogonally blocked complete diallel crosses for estimating general combining abilities is investigated when the model also includes specific combining abilities. It is proved that these designs remain optimal even in the presence of specific combining abilities. Three new series of orthogonally blocked designs are also reported. [Copyright &y& Elsevier]

Abstract: This paper describes the construction of search designs which permit the estimation of the general mean and main-effects, and allow the search for and estimation of one possibly unknown non-zero effect among the two-and three-factor interactions in the general symmetric and asymmetric factorial set-up.

This paper is in connection with search designs which permit the estimation of the general mean and main-effects, and allow the search and estimation of the possibly unknown non-zero effect among the two-factor interactions in the general asymmetric factorial set-up. This paper describes the construction of search designs in the above set-up.

Abstract: This paper considers -optimal approximate designs for random coefficient regression models. An equivalence theorem for the -optimality is provided for random coefficient regression models. Explicit expressions for -optimal designs under linear regression models are presented for illustration. [Copyright &y& Elsevier]

Abstract: A criterion for comparing two competitive designs is proposed for symmetric factorial experiment based on the probability of correct identification of active factors. An algorithm for constructing multi-level supersaturated designs which maximize the probability of correct model identification is presented. [Copyright &y& Elsevier]

In this paper we investigate geometric characterization of optimal designs for random coefficient regression models. We establish a generalization of Elving's theorem for the D -optimality criterion for estimating the population parameters in random coefficient regression models. We also consider the geometric characterization of D -optimal designs for predicting the individual parameters. [ABSTRACT FROM AUTHOR]

Fractional factorial designs are popularly used in different fields of experimentations, which allow experimenters to study large number of potentially relevant factors with relatively small number of experimental units but suffer from the fact that some of the effects are aliased with some others. In order to overcome these drawbacks, the foldover technique is widely used to de-alias factor effects. This article aims at using the mirror image reflection to augment a given design, to improve its properties in terms of the uniformity criterion measured by Lee discrepancy. [ABSTRACT FROM AUTHOR]