A characterization of some {3v2 + v3, 3v1 + v2; 3, 3}-minihypers and some [15, 4, 9; 3]-codes with B2 = 0

It is known (cf. Hill and Newton (Ars Combin., 25A (1988), 61–72; Designs Codes Cryptography, 2 (1992), 137–157) and Remark A.2 in the Appendix) that (1) there is no [14, 4, 9; 3]-code meeting the Griesmer bound and (2) if C is a [15, 4, 9; 3]-code then B₂ = 0 or 2 and (3) there is a one-to-one correspondence between the set of all nonequivalent [15, 4, 9; 3]-codes with B₂ = 0 and the set of all {3v₂ + v₃, 3v₁ + v₂: 3, 3}-minihypers, where v₁ = 1, v₂ = 4, v₃ = 13 and B₂ denotes the number of codewords of weight 2 in its dual code. Recently it has been shown by Eupen and Hill (Designs Codes Cryptography, 4 (1994) 271–282) that a [15, 4, 9; 3]-code with B₂ = 2 is unique up to equivalence. The purpose of this paper is to characterize all [15, 4, 9; 3]-codes with B₂ = 0 using the geometrical structure of the corresponding {3v₂ + v₃, 3v₁ + v₂; 3, 3}-minihypers. Those results give a complete characterization of [15, 4, 9; 3]-codes.     

Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

Self-dual codes which are principal ideals of the group algebra F2[{F2m, + }]

This paper follows up Camion's contribution on self-dual codes which are principal ideals of the algebra F₂[{F_2^m, + }], the so-called H-codes. Our main result is that this class of codes does not meet the Gilbert-Varshamov bound. We obtain this result by giving an upper bound on the minimal distance of any H-code. We characterize extremal H-codes and link up their generators with certain difference sets.     

Balanced arrays of strength 4 and balanced fractional 3m factorial designs

A connection between a balanced fractional 2^m factorial design of resolution 2l + 1 and a balanced array of strength 2l with index set {μ₀, μ₁,…, μ_2l} was established by Yamamoto, Shirakura and Kuwada (1975). The main purpose of this paper is to give a connection between a balanced fractional 3^m factorial design of resolution V and a balanced array of strength 4, size N, m constraints, 3 levels and index set {λ_l0l1l2}.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

On the spectrum of [v,k, t] trades

A [v, k, t] trade of volume m consists of two disjoint collections T₁ and T₂, each of m k-subsets of a v-set V, such that each t-subset of V is contained in the same number of blocks of T₁ and T₂, and each element of V is contained in at least one block of T₁. We study [v, k, t] trades, and investigate their spectrum (i.e., the collections of allowable volumes), using both theoretical techniques and computer-based searches.     

Optimal designs,supplementary difference sets and multipliers

We investigate multipliers of 2 - {v; q², q²; λ} supplementary difference sets where cyclotomy has been used to construct D-optimal designs.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

Perfect octagon quadrangle systems with upper C4-systems

An octagon quadrangle is the graph consisting of an 8-cycle (x₁, x₂,…, x₈) with two additional chords: the edges {x₁, x₄} and {x₅, x₈}. An octagon quadrangle system of order v and index ρ [OQS] is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of ρK_v defined on X. An octagon quadrangle systemΣ=(X,H) of order v and index λ is said to be upper C₄-perfect if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v and also the collection of all of the outside8-cycles contained in the octagon quadrangles form a ?-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems.     

Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

On some joint distributions in fluctuation theory

For non-negative integral valued interchangeable random variables v₁, v₂,…,v_n, Takács (1967, 70) has derived the distributions of the statistics ?_n' ?¹_n' ?^(c)_n and ?^(-c)_n concerning the partial sums N_r = v₁ + v₂ + ··· + v_rr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α^(c)_n, δ^(c)_n, Z_n), (β^(c)_n, Z_n) and (β^(-c)_n, Z_n) concerning the partial sums N_r = ε₁ + ··· + ε_rr = 1,2,…,n, of geometric random variables ε₁, ε₂,…,ε_n.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

An interesting property of the Poisson operating characteristic function

In elementary probability theory, as a result of a limiting process the probabilities of aBi(n, p) binomial distribution are approximated by the probabilities of aPo(np) Poisson distribution. Accordingly, in statistical quality control the binomial operating characteristic function \(\mathcal{L}_{n,c} (p)\) is approximated by the Poisson operating characteristic function \(\mathcal{F}_{n,c} (p)\) . The inequality \(\mathcal{L}_{n + 1,c + 1} (p) > \mathcal{L}_{n,c} (p)\) forp∈(0;1) is evident from the interpretation of \(\mathcal{L}_{n + 1,c + 1} (p)\) , \(\mathcal{L}_{n,c} (p)\) as probabilities of accepting a lot. It is shown that the Poisson approximation \(\mathcal{F}_{n,c} (p)\) preserves this essential feature of the binomial operating characteristic function, i.e. that an analogous inequality holds for the Poisson operating characteristic function, too.     

Linear models that allow perfect estimation

The general Gauss–Markov model, Y = Xβ + e, E(e) = 0, Cov(e) = σ ² V, has been intensively studied and widely used. Most studies consider covariance matrices V that are nonsingular but we focus on the most difficult case wherein C(X), the column space of X, is not contained in C(V). This forces V to be singular. Under this condition there exist nontrivial linear functions of Q′Xβ that are known with probability 1 (perfectly) where ${C(Q)=C(V)^\perp}$ . To treat ${C(X) \not \subset C(V)}$ , much of the existing literature obtains estimates and tests by replacing V with a pseudo-covariance matrix T = V + XUX′ for some nonnegative definite U such that ${C(X) \subset C(T)}$ , see Christensen (Plane answers to complex questions: the theory of linear models, 2002, Chap. 10). We find it more intuitive to first eliminate what is known about Xβ and then to adjust X while keeping V unchanged. We show that we can decompose β into the sum of two orthogonal parts, β = β ₀ + β ₁, where β ₀ is known. We also show that the unknown component of X β is ${X\beta_1 \equiv \tilde{X} \gamma}$ , where ${C(\tilde{X})=C(X)\cap C(V)}$ . We replace the original model with ${Y-X\beta_0=\tilde{X}\gamma+e}$ , E(e) = 0, ${Cov(e)=\sigma^2V}$ and perform estimation and tests under this new model for which the simplifying assumption ${C(\tilde{X}) \subset C(V)}$ holds. This allows us to focus on the part of that parameters that are not known perfectly. We show that this method provides the usual estimates and tests.     

L models and multiple regressions designs

Given an orthogonal model

${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$

an L model

${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$

is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L ₁, . . . , L _c ). In multiple regression designs this matrix will be of the form

${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$

with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.

     

E-optimal incomplete block designs with two distinct block sizes

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b₁,b₂,k₁,k₂) of incomplete block designs for v treatments in b₁ blocks of size k₁ each and b₂ blocks of size k₂ each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by augmenting b₂ blocks, with k₂ − k₁ extra plots each, of a BIBD(v,b = b₁ + b₂,k₁,λ) and GDD(v,b = b₁ + b₂,k₁,λ₁,λ₂). It is also shown that equireplicate E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by combining disjoint blocks of BIBD(v,b,k₁,λ) and GDD(v,b,k₁,λ₁,λ₂) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.