A note on theE-optimality of block designs

Summary In this paper we consider the problem of determiningE-optimal block designs for experimental situations in whichv treatments are to be tested onn experimental units arranged inb blocks and where the block sizes and number of replications assigned to the treatments are allowed to vary. Some sufficient conditions are obtained for designs to beE-optimal in these situations and methods for constructing designs which satisfy the sufficient conditions given are also derived.     

Six families of efficient resolvable designs in three replicates

A resolvable incomplete-block design in three replicates is abstractly equivalent to a design for three factors whose effects are additive, but the efficiency factors for the two designs are not the same. Six families of known designs of the second type are used to construct designs of the first type.     

Designs for smoothing spline ANOVA models

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2^d factorial designs, and the glp designs is given by numerical computation. Received May 2000     

Almost saturatedD-optimal main effect plans and allied results

The theory of unimodular matrices has been applied to deriveD-optimal main effect plans fors ₁ ×s ₂ factorials ins ₁ +s ₂ runs. Plans with highA-efficiency have also been given.     

Two-level search design for main-effect plus two plan

One important class of screening designs is the search design first proposed by Srivastava (1975). A new class of two-level factorial search designs which are capable of estimating all main-effect plus two interactions is provided. We first give a necessary and sufficient condition for the main-effect plus two plan and then show that the proposed search design always satisfies such a condition. Received October 2000     

DS-optimal designs in one way ANOVA

Characterization and construction of optimal designs using the familiar optimality criteria, for example A-, D- and E-optimality are well studied in the literature. However the study of the Distance Optimality (DS-) criterion introduced by Sinha (1970) has very recently drawn attention of researchers. In the present article, we consider the singularly estimable full rank problem of estimating the full set of elementary treatment contrasts using the DS optimality criterion in the set up of a one way ANOVA model. Using a limit argument it turns out that a CRD in which difference between any two allocation numbers is at the most unity is uniquely DS-optimal. Acknowledgement. We are thankful to Prof. B. K. Sinha for suggesting the problem to us and many helpful discussions with him. We are also thankful to the referees for drawing our attention to the reference of Bischoff (1995) and many helpful comments.     

Optimality and efficiency of incomplete split-block designs

In this paper, a balanced incomplete split-block design (BISBD) is shown to be universally optimum for the estimation of the interaction effects with respect to each of the intra-plot stratum estimation and the GLSE. Furthermore, the efficiency factor of an optimal design is investigated.     

Small asymmetric fractional factorial plans for main effects and specified two-factor interactions

Fractional factorial plans represented by orthogonal arrays of strength two are known to be optimal in a very strong sense under a model that includes the mean and all the main effects, when all interactions are assumed to be absent. When a fractional factorial plan given by an orthogonal array of strength two is not saturated, one might think of entertaining some two-factor interactions also in the model. In such a situation, it is of interest to examine which of the two-factor interactions can be estimated via a plan represented by an orthogonal array, as also to study the overall efficiency of the plan when some interactions are in the model alongwith the mean and all main effects. In this paper, an attempt has been made to examine these issues by considering some practically useful plans for asymmetric (mixed level) factorials with small number of runs.     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

On the choice of support of Re-descendingψ-Functions in linear models with asymmetric error distributions

InM-estimation of the regression parameter vector in the linear model, we discuss the choice of the support of certain re-descendingψ-functions for both cases when the distribution of the i.i.d. errors is partially known and when it is completely functionally unknown. Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Optimal criteria and equivalence for nonregular fractional factorial designs

By considering the so-called algebraic orthogonality among experimental runs, this paper introduces a new criterion, minimum inner-product moment (MIPM), for general asymmetrical designs, and shows that MIPM is equivalent to the minimum moment aberration (MMA) criterion for the natural weights. Furthermore, the relationship between the generalized minimum aberration (GMA) and some model-dependent efficiency criteria is investigated by using the complex contrasts. Thus, two new justifications of GMA criterion is given from the points of view of orthogonality among experimental runs and design efficiency. They are generalizations of the related results of Butler (2003) and Cheng, Deng, and Tang (2002) for two-level factorial designs.     

Utility maximization under a shortfall risk constraint

The article analyzes optimal portfolio choice of utility maximizing agents in a general continuous-time financial market model under a joint budget and downside risk constraint. The risk constraint is given in terms of a class of convex risk measures. We do not impose any specific assumptions on the price processes of the underlying assets. We analyze under which circumstances the risk constraint is binding. We provide a closed-form solution to the optimization problem in a general semimartingale framework. For a complete market, the wealth maximization problem is equivalent to a dynamic portfolio optimization problem.     

Randomized response technique for two sensitive characteristics at the same time

In this paper we present a randomized response technique which can be used to estimate the proportion of individuals having two sensitive characteristics at the same time. The information obtained from this procedure can be used for weighting (post-stratification) purposes.     

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.     

U-statistics for sequential change detection

Large sample approximations for sequential U-statistics using anti-symmetric kernels are considered both under no-change and change hypotheses. These new approximations make it easy to perform sequential tests. Received: November 1999     

Distributions of exact tests in the exponential family

The aim of this paper is to give some results on the exact density of the I-divergence in the exponential family with gamma distributed observations. It is shown in particular that the I-divergence can be decomposed as a sum of two independent variables with known distributions. Since the considered I-divergence is related to the likelihood ratio statistics, we apply the method to compute the exact distribution of the likelihood ratio tests and discuss the optimality of such exact tests. One of these tests is the exact LR test of the model which is asymptotically optimal in the Bahadur sense. Numerical examples are provided to illustrate the methods discussed. Received: January 2002 Acknowledgements. I am grateful to Prof. Andrej Pázman for helpful discussions during the setup and the preparation of the paper and to the referees for constructive comments on earlier versions of the paper. Research is supported by the VEGA grant (Slovak Grant Agency) No 1/7295/20