E-optimality of some row and column designs

In this paper we consider experimental settings where treatments are being tested in b ₁ rows and b ₂ columns of sizes k _1i and k _2j , respectively, i=1,2,..., b ₁, j=1,2,..., b ₂. Some sufficient conditions for designs to be E-optimal in these classes are derived and some necessary and sufficient conditions for the E-optimality of some special classes of row and column designs are presented. Examples are also given to illustrate this theory.     

Some E and MV-optimal designs for the two-way elimination of heterogeneity

Summary It is well known that in experimental settings wherev treatments are being tested inb blocks of sizek, a group divisible design having parametersλ ₂=λ₂+1 and whose correspondingC-matrix has maximal trace is both E and MV-optimal among all possible competing designs. In this paper, we show that under certain conditions, the E and MV-optimal group divisible block designs mentioned in the previous sentence can be used to construct E and MV-optimal row-column designs to handle experimental situations in which heterogeneity is to be eliminated in two directions and wherev treatments are being tested inb columns andk rows. Examples are given to illustrate how the results obtained can be applied. Research sponsored in part by National Science Foundation Grant No. DMS-8401943.     

Some sufficient conditions for the E- and MV-optimality of block designs having blocks of unequal size

Summary In this paper we consider experimental situations in which ν treatments are to be tested inb blocks whereb _i blocks containk _i experimental units,i=1,...,p, k ₁<k ₂<...<k _p . The idea of a group divisible (GD) design is extended to that of a group divisible design with unequal block sizes (GDUB design) and then a number of results concerning the E- and MV-optimality of GD designs are generalized to the case of GDUB designs.     

Uniformly More Powerful, Two-Sided Tests for Hypotheses about Linear Inequalities

Let Xhave a multivariate, p-dimensional normal distribution (p 2) with unknown mean and known, nonsingular covariance . Consider testing H ₀ : b _i 0, for some i = 1,..., k, and b _i 0, for some i = 1,..., k, versus H ₁ : b _i < 0, for all i = 1,..., k, or b _i < 0, for all i = 1,..., k, where b ₁,..., b _k , k 2, are known vectors that define the hypotheses and suppose that for each i = 1,..., k there is an j {1,..., k} (j will depend on i) such that b _i b _j 0. For any 0 < < 1/2. We construct a test that has the same size as the likelihood ratio test (LRT) and is uniformly more powerful than the LRT. The proposed test is an intersection-union test. We apply the result to compare linear regression functions.     

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

In this study, we have found upper and lower bounds for the spectral norm of Cauchy–Toeplitz and Cauchy–Hankel matrices in the forms T_n=[1/(a+(i−j)b)]ⁿ_i,j=1, H_n=[1/(a+(i+j)b)]ⁿ_i,j=1.     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

Average worth and simultaneous estimation of the selected subset

Suppose a subset of populations is selected from the given k gamma G( _i,p ) (i = 1,2,...,k)populations, using Gupta's rule (1963, Ann. Inst. Statist. Math., 14, 199–216). The problem of estimating the average worth of the selected subset is first considered. The natural estimator is shown to be positively biased and the UMVUE is obtained using Robbins' UV method of estimation (1988, Statistical Decision Theory and Related Topics IV, Vol. 1 (eds. S. S. Gupta and J. O. Berger), 265–270, Springer, New York). A class of estimators that dominate the natural estimator for an arbitrary k is derived. Similar results are observed for the simultaneous estimation of the selected subset.     

Packing Arrays and Packing Designs

A packing array is a b × k array, A with entriesa _i,j from a g-ary alphabet such that given any two columns,i and j, and for all ordered pairs of elements from a g-ary alphabet,(g ₁, g ₂), there is at most one row, r, such thata _r,i = g ₁ anda _r,j = g ₂. Further, there is a set of at leastn rows that pairwise differ in each column: they are disjoint. A central question is to determine, forgiven g, k and n, the maximum possible b. We examine the implications whenn is close to g. We give a brief analysis of the case n = g and showthat 2g rows is always achievable whenever more than g exist. We give an upper bound derivedfrom design packing numbers when n = g – 1. When g + 1 k then this bound is always at least as good as the modified Plotkin bound of [12]. When theassociated packing has as many points as blocks and has reasonably uniform replication numbers, we show thatthis bound is tight. In particular, finite geometries imply the existence of a family of optimal or near optimalpacking arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when .     

Optimum experimental design for a regression on a hypercube-generalization of Hoel's result

In the note Hoel's result (1965, Ann. Math. Statist., 36, 1097–1106) is generalized to a large family of experimental design optimality criterions. Sufficient conditions for optimality criterion are given, which ensure existence of the optimum experimental design measure which is a product of design measures on lower dimensional domains.     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

Old and new designs via difference multisets and strong difference families

Let X =((x_1,1,x_1,2,…,x_1,k),(x_2,1,x_2,2,…,x_2,k),…,(x_t,1,x_t,2,…,x_t,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (x_h,i-x_h,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.     

On a Turán type problem of Erdös

Let L ^k be the graph formed by the lowest three levels of the Boolean lattice B _k , i.e.,V(L ^k )={0, 1,...,k, 12, 13,..., (k–1)k} and 0is connected toi for all 1ik, andij is connected toi andj (1i<jk).It is proved that if a graph G overn vertices has at leastk ^3/2 n ^3/2 edges, then it contains a copy of L ^k .Research supported in part by the Hungarian National Science Foundation under Grant No. 1812     

Mixed block designs

Let V_i (i = 1, 2) be a set of size v_i. Let D be a collection of ordered pairs (b₁, b₂) where b_i is a k_i-element subset of V_i. We say that D is a mixed t-design if there exist constants λ _(j,j2), (0 ≤ j_i ≤ k_i, j₁ + j₂ ≤ t) such that, for every choice of a j₁-element subset S₁ of V₁ and every choice of a j₂-element subset S₂ of V₂, there exist exactly λ_(j1,j2) ordered pairs (b₁, b₂) in D satisfying S₁ ⊆ b₁ and S₂ ⊆ b₂. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v₁ + v₂ − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v₁ + v₂ − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998     

On computing the dispersion function

A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF ₁,...,F _n and a probabilityp, find an interval [a,b] of minimum width such thatF _i(b)–F _i(a ^–)p, fori=1,...,n.The author wishes to thank Dr. I. D. Coope, for helpful advice offered during the preparation of this paper, and the referee, whose comments contributed to a clearer presentation.     

Selection in Monotone Matrices and Computingkth Nearest Neighbors

Anm×nmatrix =(a_{i, j}), 1≤i≤mand 1≤j≤n, is called atotally monotonematrix if for alli₁, i₂, j₁, j₂, satisfying 1≤i₁<i₂≤m, 1≤j₁<j₂≤n.[formula]We present an[formula]time algorithm to select thekth smallest item from anm×ntotally monotone matrix for anyk≤mn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for ageneralized row-selection problemin monotone matrices. Given a setS={p₁,…, p_n} ofnpoints in convex position and a vectork={k₁,…, k_n}, we also present anO(n^4/3log^c n) algorithm to compute thek_ith nearest neighbor ofp_ifor everyi≤n; herecis an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points ofSare arbitrary, then thek_ith nearest neighbor ofp_i, for alli≤n, can be computed in timeO(n^7/5 log^c n), which also improves upon the previously best-known result.     

The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let B_k denote the subset of B determined by the designs with block size k. We present some information about B and the sets B_k. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k^′ such that the equation b^′=b+h has infinitely many solutions b,b^′ satisfying bB_k and b^′B_k^′. The study is restricted to the case λ=1.     

Remarks on band matrices

In this note we consider band- or tridiagonal-matrices of orderk whose elements above, on, and below the diagonal are denoted byb _i ,a _i,c _i . In the periodic case, i.e.b _i+m =b _i etc., we derive fork=nm andk=nm–1 formulas for the characteristic polynomial and the eigenvectors under the assumption that i=1 In the latter case it is shown that the characteristic polynomial is divisible by them–1-th minor, as was already observed byRósa. We also give estimations for the number of real roots and an application to Fibonacci numbers.     

Conservative loads on shells

Generalized load vectorsp and edge load vectorsF are denned in terms of the body force and surface on a shell. Necessary and sufficient conditions are derived forp andF, and therefore the body force and the surface force, to be conservative. It is shown for example thatp must satisfyp _i=P _ijk q _j,1 q _k,2+Q _ij ¹ q _j,2–Q _ij ² q _j,1+R _i whereq is the generalized position vector andP _ijk, Q_i,j ¹ and Q_ij ² are skew tensors.The case of hydrostatic pressure is examined in detail.This work was supported in part by NSF grant MSM 8618657.     

Fast formula of a reliability ofm-consecutive-k-out-of-n:F system with cyclek

Anm-consecutive-k-out-of-n:F system is a system ofn linearly arranged components which fails if and only if at leastm non-overlapping sequences ofk components fail, when there arek distinct components with failure probabilitiesq _i fori=1,...,k and where the failure probability of thej-th component (j=rk+i (1 ≤i ≤k) isq _j =q _i , we call this system by anm-consecutive-k-out-of-n:F system with cycle (or period)k. In this paper we give a formula of the failure probability ofm-consecutive-k-out-of-n:F system with cyclek via the failure probability of consecutive-k-out-of-n:F system.     

A remark on the intersection arrays of distance regular graphs and the distance regular graphs of diameter d = 3i − 1 with b i = 1 and k > 2

Let be a distance regular graph with intersection array b ₀, b ₁,..., b _d–1; c ₁,..., c _d. It is shown that in some cases (c _i–1, a _i–1, b _i–1) = (c ₁, a ₁, b ₁)and (c _2i–1, a _2i–1, b _2i–1) imply k 2b _i + 1. As a corollary all distance regular graphs of diameter d = 3i – 1 with b _i = 1 and k > 2 are determined.