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.     

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 .     

Two-stage DEA: <Emphasis Type=Italic>caveat emptor</Emphasis>

This paper examines the wide-spread practice where data envelopment analysis (DEA) efficiency estimates are regressed on some environmental variables in a second-stage analysis. In the literature, only two statistical models have been proposed in which second-stage regressions are well-defined and meaningful. In the model considered by Simar and Wilson (J Prod Anal 13:49–78, 2007), truncated regression provides consistent estimation in the second stage, where as in the model proposed by Banker and Natarajan (Oper Res 56: 48–58, 2008a), ordinary least squares (OLS) provides consistent estimation. This paper examines, compares, and contrasts the very different assumptions underlying these two models, and makes clear that second-stage OLS estimation is consistent only under very peculiar and unusual assumptions on the data-generating process that limit its applicability. In addition, we show that in either case, bootstrap methods provide the only feasible means for inference in the second stage. We also comment on ad hoc specifications of second-stage regression equations that ignore the part of the data-generating process that yields data used to obtain the initial DEA estimates.     

Inference by the <Emphasis Type=Italic>m</Emphasis> out of <Emphasis Type=Italic>n</Emphasis> bootstrap in nonparametric frontier models

It is well-known that the naive bootstrap yields inconsistent inference in the context of data envelopment analysis (DEA) or free disposal hull (FDH) estimators in nonparametric frontier models. For inference about efficiency of a single, fixed point, drawing bootstrap pseudo-samples of size m < n provides consistent inference, although coverages are quite sensitive to the choice of subsample size m. We provide a probabilistic framework in which these methods are shown to valid for statistics comprised of functions of DEA or FDH estimators. We examine a simple, data-based rule for selecting m suggested by Politis et al. (Stat Sin 11:1105–1124, 2001), and provide Monte Carlo evidence on the size and power of our tests. Our methods (i) allow for heterogeneity in the inefficiency process, and unlike previous methods, (ii) do not require multivariate kernel smoothing, and (iii) avoid the need for solutions of intermediate linear programs.     

Post-<Emphasis Type=Italic>Burlington</Emphasis>: What Employers and Employees Need to Know About Retaliation

Since the landmark decision in Burlington Northern & Santa Fe Railway Co. v. White, numerous federal district courts and circuit courts of appeals in the United States have considered employees’ retaliation claims. This paper reviews several post-Burlington cases and provides employers with a roadmap as to what has been held to be retaliation under the law and what has not. Our contribution is an up-to-date analysis of retaliation cases, on a specific employment action basis, to provide guidance to employers of the types of activity that could support a claim of retaliation and to employees to alert them as to the types of activity that they should not have to endure in the workplace.     

<Emphasis Type="Italic">Notes and Comments:</Emphasis> Profitability in a multiple strategy market

The link between martingales and arbitrage is well-known in financial theory: arbitrage is not available if and only if there exists an equivalent measure such that the discounted prices are martingales with respect to this measure (MME). As a consequence, under MME, any previsible (non-anticipative) strategy cannot have secure (without risk) profit. Moreover, a careful reading of a bootstrap proof of the first fundamental theorem of asset pricing (see Schachermeier (1992)) underlines the fact that, if there is no possibility of arbitrage during any unit interval, then no arbitrage is allowed with any finite temporal horizon strategy. Mathematics Subject Classification (2000): Primary: 60G48; Secondary: 60G40, 60G07 Journal of Economic Literature Classification: C50, C72, D84     

Maximal Rank Minimum Aberration Foldover Plans for 2<Superscript><Emphasis Type="Italic">m</Emphasis>-<Emphasis Type="Italic">k</Emphasis></Superscript> Fractional Factorial Designs

Two level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. In a recent paper, Li and Lin (2003) suggested a strategy for constructing optimal follow up designs using the well known foldover technique and the minimum aberration criterion. In this paper, we extend the results of Li and Lin (2003) by giving an alternate technique for constructing optimal follow up designs using the foldover technique in conjunction with the maximal rank–minimum aberration criterion suggested in Jacroux (2003).     

Shrinkage estimation of <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">X</Emphasis><<Emphasis Type="Italic">Y</Emphasis>) in the exponential case with common location parameter

We consider the problem of estimating R=P(X<Y) where X and Y have independent exponential distributions with parameters and respectively and a common location parameter . Assuming that there is a prior guess or estimate R₀, we develop various shrinkage estimators of R that incorporate this prior information. The performance of the new estimators is investigated and compared with the maximum likelihood estimator using Monte Carlo methods. It is found that some of these estimators are very successful in taking advantage of the prior estimate available.Acknowledgments. The authors are grateful to the editor and to the referees for their constructive comments that resulted in a substantial improvement of the paper.     

<Emphasis Type="Italic">Notes and Comments:</Emphasis> The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process

In continuous time, we study a financial market which is free of arbitrage opportunities but incomplete under the physical probability measure P. Thus one has several choices of equivalent martingale measures. In the present paper, the (unique) martingale measure P ^* is studied which is defined by the concept of the numeraire portfolio. The choice of P ^* can be justified by a change of numeraire in place of a change of measure. Mathematics Subject Classification (2000): 90A09, 91B28, 91B62, 93E20, 62P05 Journal of Economic Literature Classification: G10, G12, G13     

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.      

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.     

Addressing the Growing Inadequacies of the <Emphasis Type="Italic">Ellerth</Emphasis>/<Emphasis Type="Italic">Faragher</Emphasis> Affirmative Defense: Fashioning a Sensible and Feasible Solution

Public policy seeks an equitable balance between the rights and duties of both employers and employees in creating a workplace free of sexual harassment. This goal is particularly difficult to achieve when supervisors create a hostile work environment. The U.S. Supreme Courts decisions in Ellerth and Faragher created an affirmative defense against vicarious liability for employers, but subsequent applications of these rulings have been inconsistent and problematic. Courts and legislators need to (1) choose the appropriate standard of employer liability for sexual harassment committed by supervisors, and (2) decide what, if any, defenses are available to employers when employees bring claims of sexual harassment. We conclude that holding employers strictly liable while allowing for the limitation of damages based on avoidable consequences, as recently adopted by the California Supreme Court in McGinnis, is preferable to other liability standards and defenses.     

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.     

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.