A. M. Elsawah

A new look on optimal foldover plans in terms of uniformity criteria

ABSTRACT In this paper, we develop a new mechanism for finding the optimal foldover plans (OFPs) which is based on the uniformity criteria measured by Lee discrepancy, wrap-around L2-discrepancy, and centered L2-discrepancy. For three-level fractional factorials as the original designs, general foldover plans and combined designs are defined, and lower bounds of these three discrepancies of combined designs under general foldover plans are also obtained, which can be used as benchmarks for searching OFPs. Illustrative examples with a comparison study between the foldover plans under these discrepancies are provided. Our results provide a theoretical justification for OFPs of three-level designs in terms of uniformity criteria.

Asymmetric uniform designs based on mixture discrepancy

ABSTRACT Efficient experimental design is crucial in the study of scientific problems. The uniform design is one of the most widely used approaches. The discrepancies have played an important role in quasi-Monte Carlo methods and uniform design. Zhou et al. [17] proposed a new type of discrepancy, mixture discrepancy (), and showed that may be a better uniformity measure than other discrepancies. In this paper, we discuss in depth the as the uniformity measure for asymmetric mixed two and three levels U-type designs. New analytical expression based on row distance and new lower bound of the are given for asymmetric levels designs. Using the new formulation and the new lower bound as the benchmark, we can implement a new version of the fast local search heuristic threshold accepting. By this search heuristic, we can obtain mixed two and three levels U-type designs with low discrepancy.

Asymptotic Distributions of the Generalized Range, Midrange, Extremal Quotient, and Extremal Product, with a Comparison Study

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given.

A closer look at de-aliasing effects using an efficient foldover technique

ABSTRACT Foldover techniques are used to reduce the confounding when some important effects (usually lower order effects) cannot be estimated independently. This article develops an efficient foldover mechanism for symmetric or asymmetric designs, whether regular or nonregular. In this paper, we take the uniformity criteria (UC) as the optimality measures to construct the optimal combined designs (initial design plus its corresponding foldover design) which have better capability of estimating lower order effects. The relationship between any initial design and its combined design is studied. A comparison study between the combined designs via different UC is provided. Equivalence between any combined design and its complementary combined design is investigated, which is a very useful constraint that reduce the search space. Using our results as benchmarks, we can implement a powerful algorithm for constructing optimal combined designs. Our work covers as well as gives results better than recent works of about 20 articles in the last few years as special cases. So this article is a good reference for constructing effective designs.

Effective lower bounds of wrap-around L 2 -discrepancy on three-level combined designs

How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions. This paper takes the wrap-around L2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.

Asymptotic random extremal ratio and product based on generalized order statistics and its dual

ABSTRACT The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.