A Lower Bound for the Centered <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-Discrepancy on Asymmetric Factorials and its Application

The role of uniformity measured by the centered L ₂-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L ₂-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 L ₂-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.     

Blocked-Error-R 2: A Conceptually Improved Definition of the Proportion of Explained Variance in Models Containing Loops or Correlated Residuals

Bentler and Raykov (2000, Journal of Applied Psychology 85: 125–131), and Jöreskog (1999a, http://www.ssicentral.com/lisrel/column3.htm, 1999b http://www.ssicentral. com/lisrel/column5.htm) proposed procedures for calculating R ² for dependent variables involved in loops or possessing correlated errors. This article demonstrates that Bentler and Raykov’s procedure can not be routinely interpreted as a “proportion” of explained variance, while Jöreskog’s reduced-form calculation is unnecessarily restrictive. The new blocked-error-R ² (beR ²) uses a minimal hypothetical causal intervention to resolve the variance-partitioning ambiguities created by loops and correlated errors. Hayduk (1996) discussed how stabilising feedback models – models capable of counteracting external perturbations – can result in an acceptable error variance which exceeds the variance of the dependent variable to which that error is attached. For variables included within loops, whether stabilising or not, beR ² provides the same value as Hayduk’s (1996) loop-adjusted-R ². For variables not involved in loops and not displaying correlated residuals, beR ² reports the same value as the traditional regression R ². Thus, beR ² provides a conceptualisation of the proportion of explained variance that spans both recursive and nonrecursive structural equation models. A procedure for calculating beR ² in any SEM program is provided.     

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.     

Optimal variance- and efficiency-balanced designs for one- and two-way elimination of heterogeneity

Summary In this paper, a series ofE-optimal non-binary variance balanced (block or row-column) designs and a series ofE-optimal non-binary efficiency balanced (block or row-column) designs are provided in certain broad classes of competing designs. Furthermore, their high efficiencies by the usualA- andD-optimality criteria are shown.     

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.     

Statistical inferences accounting for human behavior

While giving subjective responses, a small fraction of respondents overstate or understate their true status. The extent of over (or under) evaluation and the proportions of such respondents differs from population to population. These biases alter the significance level and power of the commonly used tests. In this paper the authors determine the effects of these biases on the commonly used Z test, F test of Anova, and χ² test for testing the equality of multinominal distributions.This paper was presented in the International Conference in Recent Advances in Survey Sampling, in honor of Professor J.N.K. Rao, held at Ottawa in July, 2002.     

Economically optimalc- andnp-control charts

In v. Collani (1981, 1986, 1987a and 1987b) simple procedures are developed to determine the approximately optimal economic design of control charts for measurements. Applying these procedures to the case of control charts for attributes, nomograms are obtained from which the approximately optimal design ofc-charts, i.e. charts for defects, is readily available. Furthermore it is shown that this method also provides good approximately optimalnp-charts, i.e. charts for defectives. Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

A relationship between efficiencies of marginal designs and the product design

We derive theD- andG-efficiencies of product designs in a multifactor experiment in terms of theD- andG-efficiencies of the designs in the marginal models. Work supported by grants Ku 719/2 and 477/645/96 of the Deutsche Forschungsgemeinschaft     

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.     

TypeS designs with nested rows and columns

Summary Supplemented balance in block designs with nested rows and columns is considered, under the name of typeS designs with nested rows and columns. Analysis and construction of such designs are also discussed.     

Nested balancedN-ary designs

Nested balancedn-ary designs are introduced. Some methods of construction of such designs are presented with some series of designs.     

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.     

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.     

Minimal efficiency of designs under the class of orthogonally invariant information criteria

Consider the linear regression model with uncorrelated errors and an experimental design . In the article, we address the problem of calculating the minimal efficiency of with respect to the class of orthogonally invariant information criteria, containing all Kiefers criteria of _p-optimality, among others. We show that the -minimal efficiency of is equal to the minimal efficiency of with respect to a finite class of criteria which generalize the criterion of E-optimality. We also formulate conditions under which a design is maximin efficient, i.e. the most efficiency-stable for criteria from . To illustrate the results, we calculated the -minimal efficiency of _p (in particular D, A and E) optimal designs for polynomial regression on [–1,1] up to degree 4. Moreover, for the quadratic model we explicitly constructed the -maximin efficient design.Acknowledgement. The author would like to thank prof. Pázman as well as an anonymous referee for useful and inspiring comments on earlier versions of this article.Supported by: VEGA grant of the Slovak Agency No. 1/0264/03     

Necessary and sufficient conditions for consistency of generalizedM-estimates

GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximateM-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximateM-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximateM-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency ofM-estimators known from the literature. A linear regression example is presented where theM-estimate is consistent and an approximateM-estimate is incosistent.Supported by CSAS grant N. 17503.     

When does defection pay?

The present paper investigates cluster adjustment to changing economic environments by focussing on the role and stability of institutional arrangements in their local culture. It postulates two idealtypical local cultures where firms act in the common (collective) or in their own interest (egoistic). By comparing adjustment performance and stability for both types, the model finds that clusters in very volatile environments are unlikely to exhibit collective local cultures as these are unstable and provide only limited benefits for adjustment performance. Clusters facing more stable environments are more likely to show collective local cultures as these increase adjustment performance and are more stable against individual defection. Both findings suggest that collective local cultures in clusters can be relatively stable for limited environment volatility.     

$${\phi_p}$$-optimal designs for a linear log contrast model for experiments with mixtures

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S ^k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these -optimal designs behave. Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001.     

Manipulation via merging and splitting in claims problems

In claims problems, we study coalitional manipulations via claims merging and splitting. We characterize (division) rules that are non-manipulable via (pairwise) splitting and that also satisfy standard axioms of equal treatment of equals, consistency, and continuity. And we obtain a similar result for non-manipulability via (pairwise) merging.Received: 22 January 2002, Accepted: 5 February 2003, JEL Classification: C70, C71, D70I am grateful to William Thomson for many helpful comments and suggestions. I also thank Murat Sertel and an anonymous referee for many helpful suggestions for the revision. All remaining errors are mine.     

On the finite sample behavior of testing whether new is better than used of a specified age

Hollander, Park and Proschan (1986) proposed a test of new is better than used of a specified age. It is based on large sample normality of the test statistic. There is, however, no study in the literature on its actual size for small and moderate sample sizes. To shed some lights on this, the results of a Monte Carlo simulation study as well as two real data examples are reported and these indicate that the test can have a quite liberal size, especially for small to moderate sample sizes. In order to improve on this weakness, a modified test is proposed and studied. It is noticed that this modified test seems to over-correct the original test to an extent that it becomes unduly conservative sometimes. Hence we propose another modification that combines the original test and the modified test turns out to have its size quite close to the nominal level and is therefore preferable to both the original and modified tests.     

Comparing means of several treatments with a control when one observation exists per treatment

In an one-way analysis of variance with standard assumptions suppose that only one observation exists per treatment. In addition, assume that one of the treatments is a control group. Because of insufficient observations, the variance of the populations cannot be estimated and hence the usual methods for comparing treatments with the control group fail. In this paper, we present a method to compare treatments with a control when one observation exits per treatment. An algorithm is given to estimate the critical values of the test. The power of the test is investigated by a Monte Carlo simulation; numerical studies show that when there is a treatment whose mean is close to the control group, the power of the test is satisfactory.