A note on the uniform distribution on the arcsin points

In his book Pukelsheim [8] pointed out that designs supported at the arcsin points are very efficient for the statistical inference in a polynomial regression model. In this note we determine the canonical moments of a class of distributions which have nearly equal weights at the arcsin points. The class contains theD-optimal arcsin support design and theD ₁-optimal design for a polynomial regression. The results allow explicit representations ofD-, andD ₁-efficiencies of these designs in all polynomial models with a degree less than the number of support points of the design.     

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.     

Efficient paired choice designs with fewer choice pairs

For paired choice experiments, two new construction methods of designs are proposed for the estimation of the main effects. In many cases, these designs require about 30–50% fewer choice pairs than the existing designs and at the same time have reasonably high D-efficiencies for the estimation of the main effects. Furthermore, as against the existing efficient designs, our designs have higher D-efficiencies for the same number of choice pairs.     

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.     

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.     

$${\phi_p}$$-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.     

SomeE andMV-optimal row-column designs having equal numbers of rows and columns

In this paper we consider experimental situations in whichv treatments are to be tested using a row-column design consisting ofb columns andb rows and wherev does not divideb ². Some sufficient conditions are obtained for a design to beE orMV-optimal in such an experimental setting and methods for constructing row-column designs satisfying the sufficient conditions obtained are also given. This research was supported by NSF Grant No. DMS-8401943.     

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 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.     

Screening properties of certain two-level designs

Two-level screening designs are appropriate for situations where a large number of factors (q) is examined but relatively few (k) of these are expected to be important. It is not knownwhich of theq factors will be the important ones, that is, it is not known whichk dimensions of the experimental space will be of further interest. After the results of the design have received a first analysis, the design will be projected into thek dimensions of interest. These projections are investigated for Plackett and Burman type-screening designs withq23 factors, andk=3, 4, and 5.     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

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 a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

Eight-run two-level factorial designs under dependence

Dependent observations commonly arise in factorial experiments. Apart from main-effects two-level designs formed by the Cheng & Steinberg reverse foldover algorithm, which are known to be very efficient designs under dependence using the D-criterion, little is known about other designs, models and criteria, and the range of possible behaviour. In this paper, we investigate in detail 8-run two-level designs. Received February 1998     

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.     

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.     

Construction of block designs admitting an Abelian automorphism group

Construction of block designs admitting an action of a presumed automorphism group consists of two basic steps: 1.construction of orbit structures for the given automorphism group, 2. construction of block designs for the orbit structures obtained in 1. For an abelian group G, the second step, called indexing, often lasts too long, because there are too many possibilities to be enumerated with today's computer facilities. To make such a construction possible we use a principal series of the group G to obtain a refinement of the orbit structures.     

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.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.