A construction of balanced arrays of strengtht and some related incomplete block designs

Summary Saha [6] has shown the equivalence between a ‘tactical system’ (or at-design) and a 2-symbol balanced array (BA) of strengtht. The implicit method of construction of BA in that paper has been generalized herein to that of ans-symbol BA of strengtht. Some BIB and PBIB designs are also constructed from these arrays. Majindar [2], Vanstone [8] and Saha [6] have all shown that the existence of a symmetrical BIBD forv treatments implies the existence of six more BIBD's forv treatments in (v/2) blocks. An analogue of this result has been obtained for a large class of PBIB designs in this paper.     

Pairwise and variance balanced incomplete block designs

Summary The purpose of this paper is three-fold. The first purpose is to compile and to systematize published and dispersed results on two aspects of balancing in incomplete block designs, i.e., pairwise balance and variance balance. This was done in order to establish the status of these two concepts of balance in published literature and to put them in a form which is useful for further work in this area. Also, the results in this form are necessary for the development of the remainder of the paper. The second purpose of this paper is to present a method of constructing unequal replicate and/or unequal block size experiment designs for which the variance balance property is achieved. The method of construction involves the union of blocks from two or more block designs and the augmentation of some of the blocks with additional treatments; the method is denoted asunionizing block designs. A straight-forward extension of the method would produce a partially balanced block design with unequal replicate and/or unequal block designs. The enlargement of the concept and availability of variance blanced block designs to accommodate unequal replication and/or unequal block sizes is important to the researcher, the teacher, and the experimenter needing such designs. For example, an animal nutritionist or a psychologist is no longer required to have constant litter or family sizes for the blocks and may have unequal replication on the treatments for those treatments with insufficient material and still attain the goal of equal variances on all normalized treatment contracts. The third purpose of the paper is to apply the unionizing block designs method to construct a family of unequal replicate and unequal block size variance balanced designs. Some comments are given on the extension of the unionizing block designs method to construct other families of variance balanced or partially balanced block designs. This investigation was supported in part by PHS Research Grant No. R01-GM-05900 from General Medical Services.     

A hole‐size bound for incomplete t‐wise balanced designs

An incomplete t‐wise balanced design of index λ is a triple (X,H,??) where X is a υ–element set, H is a subset of X called the hole, and B is a collection of subsets of X called blocks, such that, every t‐element subset of X is either in H or in exactly λ blocks, but not both. If H is a hole in an incomplete t‐wise balanced design of order υ and index λ, then |H| ≤ υ/2 if t is odd and |H| ≤ (υ ? 1)/2 if t is even. In particular, this result establishes the validity of Kramer's conjecture that the maximal size of a block in a Steiner t‐wise balanced design is at most υ/2 if t is odd and at most (υ?1)/2 when t is even. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 269–284, 2001     

Test-Based Variable Selection via Cross-Validation

Abstract

Test-based variable selection algorithms in regression often are based on sequential comparison of test statistics to cutoff values. A predetermined a level typically is used to determine the cutoffs based on an assumed probability distribution for the test statistic. For example, backward elimination or forward stepwise involve comparisons of test statistics to prespecified t or F cutoffs in Gaussian linear regression, while a likelihood ratio. Wald, or score statistic, is typically used with standard normal or chi square cutoffs in nonlinear settings. Although such algorithms enjoy widespread use, their statistical properties are not well understood, either theoretically or empirically. Two inherent problems with these methods are that (1) as in classical hypothesis testing, the value of α is arbitrary, while (2) unlike hypothesis testing, there is no simple analog of type I error rate corresponding to application of the entire algorithm to a data set. In this article we propose a new method, backward elimination via cross-validation (BECV), for test-based variable selection in regression. It is implemented by first finding the empirical p value α*, which minimizes a cross-validation estimate of squared prediction error, then selecting the model by running backward elimination on the entire data set using α* as the nominal p value for each test. We present results of an extensive computer simulation to evaluate BECV and compare its performance to standard backward elimination and forward stepwise selection.     

Minimal balanced sets and finite geometries

A balanced set is a collection of subsets of a finite set that can be weighted so as to cover the whole set uniformly. Minimal balanced sets are of interest in the theory of n-person games, in particular for the existence of outcomes that cannot be improved upon by any coalition (core of the game).The object of this paper is to determine the finite geometries which are minimal balanced sets. We prove that the dual of any t-design with t ? 2 is a minimal balanced set. In particular symmetrical 2-designs (as projective spaces, biplanes, etc.) are always minimal balanced sets. For 1-designs the problem becomes much more difficult, but it is for instance easy to prove that any partial geometry which is not the dual of a 2-Steiner system is never a minimal balanced set; in particular generalized quadrangles are never minimal balanced sets. For linear graphs the problem is completely solved: the dual of a connected linear graph is a minimal balanced set if and only if this linear graph is not bichromatic.     

Some optimal designs for comparing a set of test treatments with a set of controls

In this paper, we give a detailed study of the problem of optimally comparing a set of t test treatments to a set of s controls under a 0-way elimination of heterogeneity model. The relationships between designs that are A and MV-optimal for comparing the test treatments to the controls and those that are A and MV-optimal for comparing all treatments are also studied.Research is sponsored by NSF Grant No. DMS-8700945.     

The statistics of words on rings

We analyze sequences of letters on a ring. Our objective is to determine the statistics of the occurrences of a set of r‐letter words when the sequence is chosen as a periodic Markov chain of order ≤ r ? 1. We first obtain a generating function for the associated probability distribution and then display its Poisson limit. For an i.i.d. letter sequence, correction terms to the Poisson limit are given. Finally, we indicate how a hidden Markov chain fits into this scheme. © 2005 Wiley Periodicals, Inc.     

A problem of combinatorial designs related to authentication codes

The strong partially balanced t-designs can be used to construct authentication codes, whose probabilities P_r of successful deception in an optimum spoofing attack of order r for r = 0, 1, …, t − 1, achieve their information-theoretic lower bounds. In this paper a new family of strong partially balanced t-designs are constructed by means of rational normal curves over finite fields. Thus based on this new partially balanced t-designs a new class of authentication codes is obtained. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 417–429, 1998     

Phase‐fitted,six‐step methods for solving x′′ = f(t,x)

An explicit, six‐step method of sixth order is presented and tuned for the numerical solution of x′′ = f(t,x). This method is explicit, hybrid, and uses two function evaluations (stages) per step. Its coefficients are varied and depend on the step size. This variance comes from the demand of the method to nullify the phase errors produced when solving the standard simple oscillator. The first and second derivative of this error vanish also. Numerical tests in a set of relevant problems illustrate the efficiency of the newly derived method.     

Bichromatic and Equichromatic Lines in ℂ2 and ℝ2

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ². For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.     

Isomorphic factorizations of complete equipartite graphs

A complete equipartite graph is a complete multipartite graph such that the color classes are equinumerous. An isomorphic factorization of a graph is a partition of the line set into disjoint isomorphic parts. It is shown that an isomorphic factorization of the complete equipartite graph into t isomorphic subgraphs exists whenever the total number of lines is divisible by t.     

The Existence of Well‐Balanced Triple Systems

A triple system is a collection of b blocks, each of size three, on a set of v points. It is j‐balanced when every two j‐sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j‐balanced for each . Well‐balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2‐balanced and 3‐balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well‐balanced triple systems are developed. Using these, the spectrum of pairs for which such a well‐balanced triple system exists is determined completely.     

On tests of hypotheses about treatment effects and treatmentX places interactions,in two heteroscedastic experiments

Summary Suppose we have two independent experiments conducted with a set of ‘t’ treatments each, at different places. This paper deals with two interesting problems of testing of hypotheses associated with these experiments. The first problem deals with the test of the equality of the respective treatment effects in the two experiments. The second problem is concerned with the testing of the equality of treatment into places interactions. Though we assume normality, the variance σ ₁ ² in one experiment is assumed different from the variance σ ₂ ² in the other experiment. When no information is available aboutR=σ ₁ ² /(σ ₁ ² +σ ₂ ² ) except that 0≦R≦1, tests known as ‘bilateral tests’ are proposed in the literature, to test the hypotheses mentioned above. This paper studies some important small sample properties of these bilateral tests. More specifically we study the probability of the first and second kind of error of these bilateral tests as a function ofR. When the two experiments have the same number of observations on each treatment, the bilateral test is shown to control the first kind of error. Fort=1,2, the level of these tests is a strictly convex function ofR and hence these tests can be very conservative. Some power properties of these tests are also obtained. Two tests which are equivalent to the bilateral tests for large sample sizes, and which are superior to the bilateral tests for small sample sizes, are obtained.     

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.     

A new infinite class of S(3, {4, 5, 7}, v)

A t‐wise balanced design ( at BD) of order v and block sizes from K , denoted by S ( t , K , v ), is a pair ( X , ??), where X is a v ‐element set and ?? is a set of subsets of X , called blocks , with the property that | B |∈ K for any B ∈?? and every t ‐element subset of X is contained in a unique block. In this article, we shall show that there is an S ( 3 , { 4 , 5 , 7 }, v ) for any positive integer v ≡ 7 ( mod12 ) with v ≠ 19 . Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:68–80, 2012     

Hitting straight lines by compound Poisson process paths

In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines. Work done in part while these authors were visiting professors at the Indian Statistical Institute, Delhi Centre, New Delhi, 110016, India. This author's investigation was supported in part by the U. S. National Science Foundation Grant No. DMS-8504319. Our coauthor and friend Prem Singh Puri died on August 12, 1989. We dedicate our contribution to this paper to his memory.     

Projective spread sets

In this paper the notion of a spread set for at-spread ofPG(2t+1,q) is generalised and it is shown that certaint-spreads ofPG(n, q) correspond to these generalised spread sets. Then a projective spread set is defined and it is shown that anyt-spread ofPG(n, q) corresponds to a projective spread set. Connections between the spread set and the projective spread set of at-spread are discussed, in particular in the case of at-spread ofPG(2t + 1,q) the spread set and the projective spread set are equivalent, giving a new and straightforward construction of a spread set. The methods developed are used to show, with the aid of a computer, that the 1-packing ofPG(7,2) constructed by Baker is regulus-free.Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday     

The bigraphical t-wise balanced designs of index 1

The t-wise balanced designs of index 1 whose point set are the edges of K_m,n, the complete bipartite graph, and that have the subgroup of all automorphisms that fix the two independent sets of K_m,n as an automorphism group are studied. All such designs are found. © 1994 John Wiley & Sons, Inc.     

Basic Properties of the (t, n)-Threshold Visual Secret Sharing Scheme with Perfect Reconstruction of Black Pixels

In this paper we consider the (t, n)-threshold visual secret sharing scheme (VSSS) in which black pixels in a secret black-white images is reproduced perfectly as black pixels when we stack arbitrary t shares. This paper provides a new characterization of the (t, n)-threshold visual secret sharing scheme with such a property (hereafter, we call such a VSSS the (t, n)-PBVSSS for short). We use an algebraic method to characterize basis matrices of the (t, n)-PBVSSS in a certain class of matrices. We show that the set of all homogeneous polynomials each element of which yields basis matrices of the (t, n)-PBVSSS becomes a set of lattice points in an (n−t+1)-dimensional linear space. In addition, we prove that the optimal basis matrices in the sense of maximizing the relative difference among all the basis matrices in the class coincides with the basis matrices given by Blundo, Bonis and De Santis [3] for all n≥ t ≥ 2.