Packing designs with equal-sized holes

The packing number, denoted by P(k,λ;g^u), is the maximum number of blocks in a λ-packing of pairs with k-tuples of a g^u-set V having u holes of size g which are disjoint and spanning. The values of P(3,λ;g^u) are determined here for all positive integers g,λ and u⩾3.     

Projective (2n,n,λ,1)-designs

A projective (2n,n,λ,1)-design is a set $\text{D}$ of n element subsets (called blocks) of a 2n-element set V having the properties that each element of V is a member of λ blocks and every two blocks have a non-empty intersection. This paper establishes existence and non-existence results for various projective (2n,n,λ,1)-designs and their subdesigns.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

Bias-corrected estimators for dispersion models with dispersion covariates

In this paper we discuss bias-corrected estimators for the regression and the dispersion parameters in an extended class of dispersion models (Jørgensen, 1997b). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the O(n⁻¹) bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results obtained by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The practical use of the formulae is that we can derive closed-form expressions for the O(n⁻¹) biases of the maximum likelihood estimators of the regression and dispersion parameters when the information matrix has a closed-form. Various expressions for the O(n⁻¹) biases are given for special models. The formulae have advantages for numerical purposes because they require only a supplementary weighted linear regression. We also compare these bias-corrected estimators with two different estimators which are also bias-free to order O(n⁻¹) that are based on bootstrap methods. These estimators are compared by simulation.     

Construction of blocked two-level regular designs with general minimum lower order confounding

Zhang et al. (2008) proposed a general minimum lower order confounding (GMC for short) criterion, which aims to select optimal factorial designs in a more elaborate and explicit manner. By extending the GMC criterion to the case of blocked designs, Wei et al. (submitted for publication) proposed a B¹-GMC criterion. The present paper gives a construction theory and obtains the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs with n≥5N/16+1$n\ge 5N/16+1$, where 2^n−m:2^r$2^{n-m}:2^r$ denotes a two-level regular blocked design with N=2^n−m$N=2^{n-m}$ runs, n   treatment factors, and 2^r$2^r$ blocks. The construction result is simple. Up to isomorphism, the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs can be constructed as follows: the n   treatment factors and the 2^r−1$2^r-1$ block effects are, respectively, assigned to the last n   columns and specific 2^r−1$2^r-1$ columns of the saturated 2^{(N−1)−(N−1−n+m)}$2^{(N-1)-(N-1-n+m)}$ design with Yates order. With such a simple structure, the B¹-GMC designs can be conveniently used in practice. Examples are included to illustrate the theory.     

Some remarks on subdesigns of symmetric designs

Let D(υ, k, λ) be a symmetric design containing a symmetric design D₁(υ₁, k₁, λ₁) (k₁ < k) and let x = υ₁(k ? k₁)/(υ ? υ₁). We show that k ≥(k₁ ? x)² + λ If equality holds, D₁ is called a tight subdesign of D. In the special case, λ₁ = λ, the inequality reduces to that of R.C. Bose and S.S. Shrikhande and tight subdesigns then correspond to their notion of Baer subdesigns. The possibilities for (7upsi;, k, λ) designs having Baer subdesigns are investigated.     

Minimax regression designs for approximately linear models with autocorrelated errors

We study the construction of regression designs, when the random errors are autocorrelated. Our model of dependence assumes that the spectral density g(ω) of the error process is of the form g(ω) = (1 − α)g₀(ω) + αg₁(ω), where g₀(ω) is uniform (corresponding to uncorrelated errors), α ϵ [0, 1) is fixed, and g₁(ω) is arbitrary. We consider regression responses which are exactly, or only approximately, linear in the parameters. Our main results are that a design which is asymptotically (minimax) optimal for uncorrelated errors retains its optimality under autocorrelation if the design points are a random sample, or a random permutation, of points from this distribution. Our results are then a partial extension of those of Wu (Ann. Statist. 9 (1981), 1168–1177), on the robustness of randomized experimental designs, to the field of regression design.     

Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S₁ and S₂. An exact N-point D  -optimal design for N=3p+q,p≥1,1≤q≤2$N=3p+q,p\ge 1,1\le q\le 2$ arranges equal weight n/N,0≤n≤p$n/N,0\le n\le p$ at the points of S₁ (S₂) and puts the remaining weight (N−3n)/N$(N-3n)/N$ on the points of S₂ (S₁) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5$N=6p+q,p\ge 1,1\le q\le 5$, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.     

General minimum lower order confounding designs: An overview and a construction theory

For fractional factorial (FF) designs, Zhang et al. (2008) introduced a new pattern for assessing regular designs, called aliased effect-number pattern (AENP), and based on the AENP, proposed a general minimum lower order confounding (denoted by GMC for short) criterion for selecting design. In this paper, we first have an overview of the existing optimality criteria of FF designs, and then propose a construction theory for 2^n−m$2^{n-m}$ GMC designs with 33N/128≤n≤5N/16$33N/128\le n\le 5N/16$, where N=2^n−m$N=2^{n-m}$ is the run size and n is the number of factors, for all N's and n  's, via the doubling theory and SOS resolution IV designs. The doubling theory is extended with a new approach. By introducing a notion of rechanged (RC) Yates order for the regular saturated design, the construction result turns out to be quite transparent: every GMC 2^n−m$2^{n-m}$ design simply consists of the last n columns of the saturated design with a specific RC Yates order. This can be very conveniently applied in practice.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

On the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.     

Fitting a multiple regression function

Consider the p-dimensional unit cube [0,1]^p, p≥1. Partition [0, 1]^p into n regions, R_1,n,…,R_n,n such that the volume Δ(R_j,n) is of order n^?1,j=1,…,n. Select and fix a point in each of these regions so that we have x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_n. Suppose that associated with the j-th predictor vector x⁽ⁿ⁾_j there is an observable variable Y⁽ⁿ⁾_j, j=1,…,n, satisfying the multiple regression model $\text{Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{=g(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{)+e}{}^{\mathrm{(n)}}{}_{\mathrm{j}}$, where g is an unknown function defined on [0, 1]^pand {e⁽ⁿ⁾_j} are independent identically distributed random variables with Ee⁽ⁿ⁾₁=0 and Var e⁽ⁿ⁾₁=σ²<∞. This paper proposes as an estimator of g(x), where k(u) is a known p-dimensional bounded density and {a_n} is a sequence of reals converging to 0 asn→∞. Weak and strong consistency of g_n(x) and rates of convergence are obtained. Asymptoticnormality of the estimator is established. Also proposed is $\text{σ}{}^2{}_{\mathrm{n}}\text{=n}{}^{\mathrm{?1}}\text{Σ}{}^{\mathrm{n}}{}_{\mathrm{j=1}}\text{(Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{?g}{}_{\mathrm{n}}\text{(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{))}{}^2$ as a consistent estimate of σ².     

Addition of runs to a two-level supersaturated design

The purpose of this article is to introduce a new class of extended E(s²)-optimal two level supersaturated designs obtained by adding runs to an existing E(s²)-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to E(s²) has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(s²)-optimal SSDs are also included.     

Improved WLP and GWP lower bounds based on exact integer programming

By using exact integer programming (IP) (integer programming in infinite precision) bounds on the word-length patterns (WLPs) and generalized word-length patterns (GWPs) for fractional factorial designs are improved. In the literature, bounds on WLPs are formulated as linear programming (LP) problems. Although the solutions to such problems must be integral, the optimization is performed without the integrality constraints. Two examples of this approach are bounds on the number of words of length four for resolution IV regular designs, and a lower bound for the GWP of two-level orthogonal arrays. We reformulate these optimization problems as IP problems with additional valid constraints in the literature and improve the bounds in many cases. We compare the improved bound to the enumeration results in the literature to find many cases for which our bounds are achieved. By using the constraints in our integer programs we prove that f(16λ,2,4)?9$f(16\lambda ,2,4)?9$ if λ$\lambda$ is odd where f(2^tλ,2,t)$f(2^t\lambda ,2,t)$ is the maximum n   for which an OA(N,n,2,t)$\mathrm{OA}(N,n,2,t)$ exists. We also present a theorem for constructing GMA OA(N,N/2-u,2,3)$\mathrm{OA}(N,N/2-u,2,3)$ for u=1,…,5$u=1,\dots ,5$.