Discrete discrepancy in factorial designs

Discrepancy measure can be utilized as a uniformity measure for comparing factorial designs. A so-called discrete discrepancy has been used to evaluate the uniformity of factorials. In this paper we give linkages among uniformity measured by the discrete discrepancy, generalized minimum aberration, minimum moment aberration and uniformity measured by the centered L₂-discrepancy/the wrap-around L₂-discrepancy. These close linkages provide a significant justification for the discrete discrepancy used to measure uniformity of factorial designs.     

Lower bounds of various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this paper, some lower bounds of various discrepancies of combined designs, such as centered L ₂-discrepancy, symmetric L ₂-discrepancy and wrap-around L ₂-discrepancy, under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L₂-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.     

A method for generating uniformly scattered points on the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-norm unit sphere and its applications

Tashiro (Ann Inst Stat Math 29:295–300, 1977) studied methods for generating unform points on the surface of the regular unit sphere. The L _p -norm unit sphere is a generalization of the regular unit sphere. In this paper we propose a method associated with an algorithm for generating uniformly scattered points on the L _p -norm unit sphere and discuss its applications in statistical simulation, representative points of a wide class of multivariate probability distributions and optimization problems. Some examples are illustrated for these applications. This research was supported by The University of Hong Kong Research Grant and a University of New Haven 2005 and 2006 Summer Faculty Fellowships.     

Connections between uniformity and aberration in general multi-level factorials

Discrepancy is a kind of important measure used in experimental designs. Among various existing discrepancies, the discrete discrepancy, centered L ₂-(CD ₂) and wrap-around L ₂-discrepancy (WD ₂) have been well justified and widely used. In this paper, using the second-order polynomials of indicator functions for these three discrepancies, we investigate the close relationships between them and the generalized wordlength pattern, and provide some conditions under which a design having one of these minimum discrepancies is equivalent to having generalized minimum aberration (GMA). These results provide further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrepancies in the quadratic forms of the indicator functions are useful for us to find optimal designs under any of them.     

Evaluating expectations of <Emphasis Type="Italic">L</Emphasis>-statistics by the Steffensen inequality

By combining the Moriguti and Steffensen inequalities, we obtain sharp upper bounds for the expectations of arbitrary linear combinations of order statistics from iid samples. The bounds are expressed in terms of expectations of the left truncated parent distribution and constants that depend only on the coefficients of the linear combination. We also present analogous results for dependent id samples. The bounds are especially useful for L-estimates of the scale parameter of the distribution.     

On the construction of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">s</Emphasis><Superscript>2</Superscript>)-optimal supersaturated designs

Supersaturated designs are an important class of factorial designs in which the number of factors is larger than the number of runs. These designs supply an economical method to perform and analyze industrial experiments. In this paper, we consider generalized Legendre pairs and their corresponding matrices to construct E(s ²)-optimal two-level supersaturated designs suitable for screening experiments. Also, we provide some general theorems which supply several infinite families of E(s ²)-optimal two-level supersaturated designs of various sizes.      

Bayesian Analysis in the <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Norm of the Mixing Proportion Using Discriminant Analysis

We consider the mixing proportion π in a mixture of two independent distributions, and establish the expression of its posterior density, in closed form and in terms of L ¹-norms of various related functions, using a prior beta and the optimal classification rule for the two populations provided by Discriminant analysis. A numerical example fully illustrates the concepts presented.Research partially supported by CRSNG 9249 (Canada). The authors wish to thank the Faculty of Science and the Department of Statistics of UNISA for their generous support that has led to this joint work. Also, thanks to Ms. Jeannette LeBlanc for her excellent technical support, and to an anonymous referee for very helpful comments that have helped to improve the presentation of the paper.     

Construction of D-optimal s 1 × s 2 × s 3 factorial designs using graph theory

In this paper, the maximum determinant of the associated 0-1 matrix in D-Optimal saturated main effect plans for 3× s_2 × s_3 factorials, is derived by the use of Graph theory and Combinatorics. The present work is related to a problem suggested by Chatterjee and Narasimhan (2002). Using the theoretical results, we also give the designs for s₃≥s₂ + 1. This research was supported by the State Scholarships Foundation of Greece.     

On Minimally-supported <Emphasis Type="Italic">D</Emphasis>-optimal Designs for Polynomial Regression with Log-concave Weight Function

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.     

Multi-level and mixed-level <Emphasis Type="Italic">k</Emphasis>-circulant supersaturated designs

Supersaturated designs (SSDs) constitute an important class of fractional factorial designs that could be extremely useful in factor screening experiments. Most of the existing studies have focused on balanced designs. This paper provides a new lower bound for the \(E(f_{NOD})\)-optimality measure of SSDs with general run sizes. This bound is a generalization of existing bounds since it is applicable to both balanced and unbalanced designs. Optimal multi and mixed-level, balanced and nearly balanced SSDs are constructed by applying a k-circulant type methodology. Necessary and sufficient conditions are introduced for the generator vectors, in order to pre-ensure the optimality of the constructed k-circulant SSDs. The provided lower bounds were used to measure the efficiency of the generated designs. The presented methodology leads to a number of new families of improved SSDs, providing tools for directly constructing optimal or nearly-optimal k-circulant designs by just checking the corresponding generator vector.     

A general construction of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">s</Emphasis><Superscript>2</Superscript>)-optimal supersaturated designs via supplementary difference sets

Supplementary difference sets can be used to construct a variety of combinatorial designs. In this work, we construct supplementary difference sets that provide E(s ²)-optimal, two-level supersaturated designs with the equal occurrence property. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.     

Top-<Emphasis Type="Italic">k</Emphasis>-lists

Top-k-lists are introduced as sequences of k-dimensional random vectors with ordered components being k largest observations from a sequence of independent identically distributed random variables. Such lists changing in time are natural stochastic models of ranking tables which appear in many situations in real life, when one wants to keep a track of several best results in a given field. Here we study basic properties of top-k-lists as joint distributions, conditional structures, representations, driving examples of top-k-lists from exponential and uniform distributions, asymptotics and a relation to generalized order statistics.     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Measuring process capability index <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">pm</Emphasis></Subscript> with fuzzy data

The process capability index C _pm , which considers the process variance and departure of the process mean from the target value, is important in the manufacturing industry to measure process potential and performance. This paper extends its applications to calculate the process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } of fuzzy numbers. In this paper, the α-cuts of fuzzy observations are first derived based on various values of α. The membership function of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is then constructed based on the α-cuts of fuzzy observations. An example is presented to demonstrate how the fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is interpreted. When the quality characteristic cannot be precisely determined, the proposed method provides the most possible value and spread of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} }. With crisp data, the proposed method reduces to the classical method of process capability index C _pm .     

Optimality results on orthogonal arrays plus <Emphasis Type="Italic">p</Emphasis> runs for <Emphasis Type="Italic">s</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> factorial experiments

The optimality of designs obtained by adding p runs to an orthogonal array is studied for experiments involving m factors each at s levels. The optimality criterion used here, is the Type 1 criterion due to Cheng (1978) which is an extension of Kiefer (1975) universal optimality criterion. Unlike what happens with orthogonal array plus one run designs, the behavior of designs obtained via augmentation of an orthogonal array by p runs depends on the particular runs added.     

A note on the uniform distribution on the arcsin points

In his book Pukelsheim [8] pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains theD-optimal arcsin support design and theD ₁-optimal design for a polynomial regression. The results allow explicit representations ofD-, andD ₁-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.     

The Alpha Risk of Taguchi Method for LTB Type Quality Characteristic with L<Subscript>8</Subscript>

In Taguchi’s parameter design, the significant parameter levels are found by maximising the signal-to-noise ratio of the quality characteristic. In the analysis of variance (ANOVA) of signal-to-noise ratio, the combination of column effects to better estimate error variance is referred to as pooling. Taguchi has suggested the strategy of pooling up. When using the pooling-up strategy, there will be a tendency to make the alpha mistake more often. In this paper, it is assumed that (1) the quality chartacteristic is normally distributed and (2) the mean and standard deviation of each factor level combination are equal, then the null hypothesis should be no significant factor. Thus, the alpha risk is that some factors are misidentified as significant factors. The purpose of this paper is to investigate the alpha risk of the Taguchi method for the-larger-the-better (LTB) type problem with orthogonal array, L₈, by simulation. The results show that the alpha risk is very high.     

A general construction of E(s 2)-optimal large supersaturated designs

A general method for construction of E(s ²)-optimal, two-level supersaturated designs (SSDs) with the equal occurrence property, from supplementary difference sets is introduced. It is proved that SSDs constructed in this way are E(s ²)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of E(s ²)-optimal large SSDs.     

A note on strategies for bandit problems with infinitely many arms

A bandit problem consisting of a sequence of n choices (n) from a number of infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution F on the interval [0,1] and F is continuous with F(0)=0 and F(1)=1. The goal is to investigate the asymptotic expected failure rates of k-failure strategies, and obtain a lower bound for the expected failure proportion over all strategies presented in Berry et al. (1997). We show that the asymptotic expected failure rates of k-failure strategies when 0<b1 and a lower bound can be evaluated if the limit of the ratio F(1)–F(t) versus (1–t)^b exists as t1^– for some b>0.