Robustness of Williams Square change-over designs

In literature, Williams Square change-over designs balanced for first residual effects of treatments have been proved to be universally optimal. Here, an attempt has been made to show that these designs are robust against missing of last α [≤v−1; v being the number of periods in the design for v treatments] observations from an experimental unit. Received May 2000     

Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.     

A class of non binary unequally replicatedE-optimal designs

In a situation where the given set of parameters (b, k andv) precludes the existence of any known optimal block designs, but an optimal block design is known to exist with parametersb, k andv*>v, a new design is shown to be useful. This (b, k, v) design is obtained from the (b, k, v*) optimal design by collapsing the classes of a suitable paritition of the treatment set (of the latter design) to treatments (of the former). We call the new design a quotient of the original design. Although the quotient is non binary and unequally replicated, it turns out to beE-optimal within the class of all proper and connected designs withb, k andv, provided the replication number of the optimal design we start with is not too large.     

Optimal extended complete block designs for dependent observations

Optimal designs under general dependence structures are usually difficult to specify theoretically or find algorithmically. However, they can sometimes be found for a specific dependence structure and a particular parameter value. In this paper, a class of generalized binary block designs with t treatments and b blocks of size k>t is considered. Each block consists of h consecutive complete blocks and, at the end, an incomplete block of size k−ht (if k > ht). For a suitable number of blocks, a universally optimal design is found for a first-order stationary autoregressive process with positive correlations. Optimal generalized binary designs and balanced block designs are also considered. Some constructions for a universally optimal design are described. A negative dependence parameter, and some other dependence structures, are also considered.     

$${\phi_p}$$-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.     

Criterion-robust optimal product designs for discrimination between nested response surface models

Consider the problem of discriminating between two rival response surface models and estimating parameters in the identified model. To construct designs serving for both model discrimination and parameter estimation, the M _γ-optimality criterion, which puts weight γ (0≤γ≤1) for model discrimination and 1 − γ for parameter estimation, is adopted. The corresponding M _γ-optimal product design is explicitly derived in terms of canonical moments. With the application of the maximin principle on the M _γ-efficiency of any M _γ'-optimal product design, a criterion-robust optimal product design is proposed.     

Comparison of test vs. control treatments using distance optimality criterion

We consider the problem of comparison of one test treatment (τ₀) with a set of v control treatments (τ₁, τ₂, …, τ_v) using distance optimality [DS-optimality] criterion introduced by Sinha (1970) in some treatment-connected design settings. It turns out that the nature of DS-optimal designs is quite similar to that for the usual A−, D− and E− optimality criteria. However, the optimality problem is quite complicated in most situations. First we deal with the CRD model and derive DS-optimal allocations for a given set of treatments. The results are almost identical to the A-optimal allocations for such problems. Then we consider a block design set-up and examine the nature of DS-optimal designs. In the process, we introduce the method of weighted coverage probability and maximize the resulting expression to obtain an optimal design. Received: December 1999     

Optimal designs for a mixed interference model

This paper generalizes Kunert and Martin’s (Ann Stat 28:1728–1742, 2000) method for finding optimal designs under a fixed interference model, to find optimal designs under a mixed interference model. The results are based on the properties of information matrices in fixed and mixed models given in Markiewicz (J Stat Plan Inference 59:127–137, 1997). The method is applied to find a design which is optimal for any given variances of random neighbor effects. Research partially supported by the KBN Grant Number 5 P03A 041 21.     

Bounds on expectations of small order statistics from decreasing density populations

For i > (n + 1)/2, Danielak (Statistics 37:305–324, 2003) established an optimal positive upper mean-variance bound on the expectation of ith order statistic based on the i.i.d. sample of size n from the decreasing density population. We show that the best bounds on the expected deviation of the ith order statistics from the population mean, i ≤ (n + 1)/2, expressed in more general scale units generated by pth absolute central moments with p > 1 amount to zero. We also determine the respective strictly negative bounds in the mean absolute deviation units.     

A note on the determination of A-optimal type block designs under fixed and mixed effects models

In this paper, we consider the problem of estimating a selected set of contrasts between v treatments using a block design consisting of b blocks of size k. Traditionally, the construction of A-optimal block designs for such situations has been carried out assuming a fixed effects model. In this paper, we show that A-optimal designs constructed under a fixed effects model are robust in the sense that these designs have maximal minimal efficiency when considered among all available designs and under all possible mixed effects models. AMS 1991 subject classifications: Primary 62K05; Secondary 62K10     

Streuungszerlegung für Paar-Vergleiche

Zusammenfassung Pl?ne für Paar-Vergleich Versuche werden gegeben und die Auswertung solcher Versuche wird beschrieben. Die Arbeit st?tzt sich auf das vonBock (1958) abge?nderte Modell vonThurstone undMosteller (1951a). Die Pl?ne sind so, da? sie zu unabh?ngig verteilten Fehlern führen. Zu ihrer Auswertung kann bedenkenlos die Streuungszerlegung angewendet werden. Die Konstruktion von ausgewogenen Pl?nen (allek(k−1)/2 m?glichen Paare werden verglichen) wird beschrieben und teilweise ausgewogene Pl?ne (nicht alle m?glichen Paare werden verglichen) für bis zu 10 Objekte sind tabelliert. Es wird gezeigt, da? die Auswertung solcher Versuche mit der Auswertung von Versuchen in unvollst?ndigen Bl?cken identisch ist. Zur leichteren Auswertung der teilweise ausgewogenen Versuche sind die Inversen der Koeffizientenmatrix der Normalgleichungen tabelliert. Ein Anwendungsbeispiel aus der Wollforschung wird durchgerechnet.

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Summary Designs for paired comparisons are given and their analysis is discussed. The paper is based onBock’s modification (1958) ofThurstone andMosteller’s (1951a) model. The designs lead to independently distributed errors and analysis of variance can be applied. The construction of balanced designs (in which allk(k−1)/2 possible comparisons are made) is described and partially balanced designs (in which not all possible comparisons are made) for up to 10 objects are tabulated. It is shown that the analysis of paired comparison experiments is identical to the analysis of incomplete block experiments. To simplify the analysis of the partially balanced designs the inverses of the matrices comprising the coefficients of the normal equations are tabulated. A worked example of application to a problem in wool research is given.

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Die Arbeit wurde im South African Wool Textile Research Institute, Rhodes University, Grahamstown, Südafrika, geschrieben.     

More optimal designs for diallel cross experiments

It is shown that the complement of a universally optimal design derivable from a triangular design is again universally optimal in a class of connected designs. Furthermore, some series of universally optimal designs for diallel cross experiments are provided. These observations strengthen the result on universally optimal designs by Das, Dey and Dean (1998). Received: August 1999     

A note on connections among criteria for asymmetrical factorials

Various optimal criteria have been proposed to rank asymmetrical fractional factorial designs. Among them, the generalized minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention. Recently, Liu et al. (Stat Sin 16:1285–1297, 2006) proposed the minimum χ ² criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum χ ² criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an analytic relationship between generalized minimum aberration and minimum moment aberration is obtained for asymmetrical fractional factorial designs.     

18-run nonisomorphic three level orthogonal arrays

In this paper we construct all possible orthogonal arrays OA(18,q, 3,2) with 18 runs and 3 ≤ q ≤ 7 columns and present those that are nonisomorphic. A discussion on the novelty and the superiority of many of the designs found in terms of isomorphism and generalized minimum aberration has been made.      

A note on connectedness of block designs

Summary We give a simple proof for the well known result that a block design (for an additive, fixed effects block model withv treatments inb blocks) is connected iff its ℰ-matrix has rankv-1.     

Bayesian U-type design for nonparametric response surface prediction

This paper deals with Bayesian design over U-type designs of n runs and s factors with q levels for nonparametric response surface prediction. The criterion is developed in terms of the asymptotic approach of Mitchell et al. (Ann Statist 22: 634–651, 1994) for a specific covariance kernel. An optimal design is given in approximate design theory over the all level combinations. A connection with orthogonality and aberration is established. A lower bound for the criterion is provided, and numerical results show that this lower bound is tight.     

Prediction of <Emphasis Type="Italic">k</Emphasis>-records from a general class of distributions under balanced type loss functions

We study the problem of predicting future k-records based on k-record data for a large class of distributions, which includes several well-known distributions such as: Exponential, Weibull (one parameter), Pareto, Burr type XII, among others. With both Bayesian and non-Bayesian approaches being investigated here, we pay more attention to Bayesian predictors under balanced type loss functions as introduced by Jafari Jozani et al. (Stat Probab Lett 76:773–780, 2006a). The results are presented under the balanced versions of some well-known loss functions, namely squared error loss, Varian’s linear-exponential loss and absolute error loss or L ₁ loss functions. Some of the previous results in the literatures such as Ahmadi et al. (Commun Stat Theory Methods 34:795–805, 2005), and Raqab et al. (Statistics 41:105–108, 2007) can be achieved as special cases of our results. Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged by J. Ahmadi. M. J. Jozani’s research supported partially by a grant of Statistical Research and Training Center. é. Marchand’s research supported by NSERC of Canada. A. Parsian’s research supported by a grant of the Research Council of the University of Tehran.     

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.     

Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables

Let X ₁, X ₂, ..., X _n be independent exponential random variables such that X _i has failure rate λ for i = 1, ..., p and X _j has failure rate λ^* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D _i:n (p,q) = X _i:n −X _i-1:n the ith spacing of the order statistics X _1:n ≤ X _2:n ≤ ... ≤ X _n:n , i = 1, ..., n, where X _0:n ≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D _i:n (p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings. Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.: NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).     

On bootstrapping periodic random arrays with increasing period

The aim of the paper is to determine when the periodic block bootstrap, procedure introduced by Chan et al. (Technometrics 46(2):215–224, 2004), can be applied to arrays of random variables. Formal consistency is obtained under α-mixing or m-dependence conditions together with the assumption that the length of the period tends to infinity. On the other hand, if the period is constant, inconsistency is shown. The performance of periodic block bootstrap is also compared in simulations with moving block bootstrap. It is suggested that for the case of long-period data the first method is more effective and much more stable with respect to the length of the block size.