On a heuristic analysis of highly fractionated 2 n factorial experiments

The paper deals with a method for the analysis of highly fractionated factorial designs proposed by Raghavarao and Altan in Metrika 58:185–191 (2003). We show that the method will find “active” factors with almost any set of random numbers. Once that an alias set is found active, Raghavarao and Altan claim that their method can resolve the alias structure of the design and identify which of several confounded effects is active. We show that their method cannot do that. The error in Raghavarao and Altan’s arguments lies in the fact that they treat a set of highly dependent (sometimes even identical) F-statistics as if they were independent.     

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 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.     

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.     

On exact D-optimal designs with 2 two-level factors and n autocorrelated observations

In this paper we consider the exact D-optimal designs for estimation of the unknown parameters in the two factors, each at only two-level, main effects model with autocorrelated errors. The vector of the n random errors in the observed responses is assumed to follow a first-order autoregressive model (AR(1)). The exact D-optimal designs seek the optimal combinations of the design levels as well as the optimal run orders, so that the determinant of the information matrix of BLUEs for the unknown parameters is maximized. Bora-Senta and Moyssiadis (1999) gave some conjectures about the exact D-optimal designs based on their experience of several exhaustive searches. In this paper their conjectures are partially proved to be true.Received: January 2003 / Accepted: October 2003Partially supported by the National Science Council of Taiwan, R.O.C. under grant NSC 91-2115-M-008-013.Supported in part by the National Science Council of Taiwan, R.O.C. under grant NSC 89-2118-M-110-003.     

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     

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.     

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.     

Trend stationarity in the I(2) cointegration model

A vector autoregressive model for I(2) processes which allows for trend-stationary components and restricts the deterministic part of the process to be at most linear is defined. A two-step statistical analysis of the model is derived. The joint test of I(1) and I(2) cointegrating ranks is shown to be asymptotically similar with respect to the drift terms and the asymptotic distribution is tabulated. The cointegrating parameters are shown to be mixed Gaussian and an application for UK monetary data illustrates the proposed analysis.     

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     

On the lowest-winning-bid and the highest-losing-bid auctions

Theoretical models of multi-unit, uniform-price auctions assume that the price is given by the highest losing bid. In practice, however, the price is usually given by the lowest winning bid. We derive the equilibrium bidding function of the lowest-winning-bid auction when there are k objects for sale and n bidders with unit demand, and prove that it converges to the bidding function of the highest-losing-bid auction if and only if the number of losers n−k gets large. When the number of losers grows large, the bidding functions converge at a linear rate and the prices in the two auctions converge in probability to the expected value of an object to the marginal winner.     

Optimising a Probabilistic Model of the Development of Play in Soccer

We present a probabilistic model based on the one developed by Hernández Mendo and Anguera (Revista de Psicologíca Social, 16(1), 71–93, 2001). Here we have tried to break down the interaction contexts that the opposing teams are able to generate and transform during the game. We are aware that a given player or team does not produce consistent behaviour in similar situations. However, a degree of uncertainty is assumed to exist regarding whether the results obtained are a specific function of the analysis used. In order to carry out this research a category system which optimized that used in the previous model was developed. This system should enable the interaction between teams to be observed within the actual play of a soccer game. A lag sequential analysis was performed on the basis of a coding of the behavioural flow. After describing the behavioural patterns obtained a probabilistic model of the development of play in soccer is proposed.     

The emission cost of international sourcing: using structural decomposition analysis to calculate the contribution of international sourcing to CO2-emission growth

The effect of changes in trade patterns, particularly increasing international sourcing, on global CO₂-emissions growth has yet to be clearly understood. In this paper, we estimate the emission cost of sourcing (ECS), which originates from replacing domestic products by imports from countries with more CO₂-intensive technologies. Using a structural decomposition analysis, we find that changes in sourcing patterns between 1995 and 2007 contribute (1) to reducing territorial emissions in high-wage countries (70% of their territorial emissions growth) and (2) to increasing territorial emissions in low-wage countries (30% of their territorial emissions increase). The net global effect, the ECS, amounts to 18% of total global CO₂-emissions growth. Our results call the climate change policies based on territorial principles into question given that they disregard that differences in emission intensities between countries contribute to raising global emissions. In contrast, policies fostering the transfer of cleaner technologies to low-wage countries decrease the ECS.