Extendable Steiner designs from finite geometries

We describe a simple procedure for constructing Steiner 2-designs with the parameters of the designs of points and lines of a finite projective or affine geometry of dimension m⩾3; the codes of many of the designs constructed in this way will contain the code of the relevant finite-geometry design (a Reed-Muller or generalized Reed-Muller code). The designs can be extended to 3-designs provided that planes in the finite-geometry design extend.     

Families of Hadamard group divisible designs

By a family of designs we mean a set of designs whose parameters can be represented as functions of an auxiliary variable t where the design will exist for infinitely many values of t. The best known family is probably the family of finite projective planes with υ = b = t² + t + 1, r = k = t + 1, and λ = 1. In some instances, notably coding theory, the existence of families is essential to provide the degree of precision required which can well vary from one coding problem to another. A natural vehicle for developing binary codes is the class of Hadamard matrices. Bush (1977) introduced the idea of constructing semi-regular designs using Hadamard matrices whereas the present study is concerned mostly with construction of regular designs using Hadamard matrices. While codes constructed from these designs are not optimal in the usual sense, it is possible that they may still have substantial value since, with different values of λ₁ and λ₂, there are different error correcting capabilities.     

Flocks and partial flocks of quadrics: a survey

If O is an ovoid of PG(3,q), then a partition of all but two points of O into q−1 disjoint ovals is called a flock of O. A partition of a nonsingular hyperbolic quadric Q⁺(3,q) into q+1 disjoint irreducible conics is called a flock of Q⁺(3,q). Further, if O is either an oval or a hyperoval of PG(2,q) and if K is the cone with vertex a point x of PG(3,q)⧹PG(2,q) and base O, then a partition of K⧹{x} into q disjoint ovals or hyperovals in the respective cases is called a flock of K. The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3,q). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q. Partial flocks which are no flocks, have applications to k-arcs of PG(2,q), to translation planes and to partial line spreads of PG(3,q). Recently, the definition and many properties of flocks of quadratic cones in PG(3,q) were generalized to partial flocks of quadratic cones with vertex a point in PG(n,q), for n⩾3 odd.     

Klingenberg structures and partial designs I: Congruence relations and solutions

We consider generalizations of projective Klingenberg and projective Hjelmslev planes, mainly (b, c)-K-structures. These are triples (φ, Π, Π′) where Π and Π′ are incidence structures and φ : Π → Π′ is an epimorphism which satisfies certain lifting axioms for double flags. The congruence relations of such triples are investigated, leading to nice factorizations of φ. Two integer invariants are associated with each congruence relation, generalizing a theorem of Kleinfeld on projective Hjelmslev planes. These invariants are completely characterized for a special class of triples: the balanced, minimally uniform neighbor cohesive (1,1)-K-structures. We show that a balanced neighbor cohesive (1,1)-K-structure Π “over” a PBIBD Π′ is again a PBIBD and compute its invariants. Several methods are given for constructing symmetric “divisible” PBIBD's on arbitrarily many classes.     

Goodness-of-fit tests for semiparametric biased sampling models

Consider an s-sample biased sampling model in which the distribution function for each of the first s−1 samples is related to the unknown distribution function G of the sth sample by a known parametric selection bias weight function. Gilbert et al. (Biometrika 86 (1999) 27) gave a procedure for semiparametric maximum likelihood estimation of the parameters in this model. In many applications, information are scarce for basing the choice of the parametric weight function(s), motivating the need for goodness-of-fit tests of the hypothesis that the weight functions are correctly specified. Cramér–von Mises-type, Anderson–Darling-type, and Kolmogorov–Smirnov-type test statistics are studied which compare discrepancies between the empirical distribution of G and the semiparametric maximum likelihood estimator of G. Finite-sample properties of the tests are evaluated with simulations and with a real example of HIV genetic sequence data.     

Generalized Bhaskar Rao designs

Generalized Bhaskar Rao designs with non-zero elements from an abelian group G are constructed. In particular this paper shows that the necessary conditions are sufficient for the existence of generalized Bhaskar Rao designs with k=3 for the following groups: ?G? is odd, G=Z^r₂, and G=Z^r₂×H where 3? ?H? and r?1. It also constructs generalized Bhaskar Rao designs with υ=k, which is equivalent to υ rows of a generalized Hadamard matrix of order n where υ?n.     

Families of regular designs based on geometries

Recently Bush and Ostrom (1979) settled most of the open questions with respect to inequivalent solutions of a class of semiregular (SR) designs which can be constructed from nets. This paper is a study of the same nature for two families of regular (R) designs derived from finite projective planes. One family presents no problems, but the other which is a ‘double’ family with two parameters is much more difficult. In fact it is here solved only for designs based on planes of orders 3, 4, 5 and 8. Certain general methods exist which are indicated, but we were unable to resolve even the case 7 using this technique.Basically we show the existence of either inequivalent solutions or show there is but one solution settling a number of open cases. In particular for the case λ₁ = 2, λ₂ = 1 we give new solutions to a number of D(2) designs or group divisible designs with two associate classes which have no repeated blocks in contrast with the published solutions which have this undesirable property for a number of applications.     

On cone representations of translation planes

Let F be a field of order q. It is known that an orthogonal array of the same order q has rank n over F if and only if it is represented as a cone cut by hyperplanes in n-dimensional space over F. Here we show that translation planes have a cone representation in (n + 1)-dimensional space over F, where n is the dimension of the plane over its kernel. If the plane is a semifield plane then the representation takes a particularly nice form. Rank 3 representations of Moulton planes are also briefly discussed.     

A generalized score confidence interval for a binomial proportion

Constructing a confidence interval for a binomial proportion is one of the most basic problems in statistics. The score interval as well as the Wilson interval with some modified forms have been broadly investigated and suggested by many statisticians. In this paper, a generalized score interval CI_G(a) is proposed by replacing the coefficient 1/4 in the score interval with parameter a. Based on analyzing and comparing various confidence intervals, we recommend the generalized score interval CI_G(0.3) for the nominal confidence levels 0.90, 0.95 and 0.99, which improves the spike phenomenon of the score interval and behaves better and computes more easily than most of other approximate intervals such as the Agresti-Coull interval and the Jeffreys interval to estimate a binomial proportion.     

The family of t-designs—part I

The family of t-designs is, without any doubt, the most important family of statistical designs. Their importance is due to their statistical optimalities, desirable symmetries for analyses and interpretations, and uses for constructing other important designs and structures such as Youden designs, generalized Youden designs, optimal fractional factorial designs, error defecting and correcting binary codes, balanced arrays, combinatorial filing systems, Hadamard matrices, finite projective and affine planes, strongly regular graphs, and so on. Research in the area of t-designs has been steadily and rapidly growing, especially during the last three decades. The number of publications in this area is in the several hundreds. Since papers on t-designs are published in a variety of journals, and because of the extensive role of these designs in design of experiments and other areas we believe it is imperative to gather these results and present them in varied form to suit diverse interests. This paper is an instance of such an attempt.     

Simple bounds on availability in a model with unknown life and repair distributions

We take a fresh look at the classic model of a device supported by a single statistically identical spare and provision for repairs, with system failure resulting whenever the currently operating unit fails before the repair of the previously failed unit is completed to allow it to become a spare. The limiting availability A(F,G) of this system depends on the life distribution F and repair time distribution G through α=∫GdF and the expected downtime. In this paper we derive several computable and sharp bounds on A(F,G) when F,G have suitable life distribution characteristics in the sense of reliability theory but are otherwise unknown except for at most two moments. Among other results, we find a sharp bound which involves the MTBF, MTTR and the second moment of the life-distribution of the device through its coefficient of variation. This leads to a maximin result for DFR repairs and DMRL lives.     

Existence of certain class of duadic group algebra codes

Duadic codes are defined in terms of idempotents of a group algebra GF(q)G, where G is a finite group and gcd(q,|G|)=1. Under the conditions of (1) q=2^m, and (2) the idempotents are taken to be central and (3) the splitting is μ₋₁, we show that such duadic codes exist if and only if q has odd-order modulo |G|.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

Bayesian estimation for the M/G/1 queue using a phase-type approximation

This article deals with Bayesian inference and prediction for M/G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase-type distributions. Given this phase-type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.     

Auxiliary sets of matrices with non-increasing step parameter sequences

The following result is proved: for every projective plane ?p there exists a projective Hjelmslev plane π of type 4 over ?p such that the step parameter sequence of π is non-increasing.     

Orthogonal [k − 1, k + 1]-factorizations in graphs

Let G be a graph. Let F={F₁,F₂,...F_d} be a factorization of G and H be a subgraph of G. If H has exactly one edge in common with F_i for all i = 1,2,…,d, then we say that F is orthogonal to H. In this paper it is proved that for any d-matching M of a [kd − 1, kd + 1]-graphG, there is a [k − 1, k + 1]-factorization of G orthogonal to M where k ⩾ 2 is an integer.     

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.     

Constrained nonparametric maximum-likelihood estimation for mixture models

A nonparametric mixture model specifies that observations arise from a mixture distribution, ∫ f(x, θ) dG(θ), where the mixing distribution G is completely unspecified. A number of algorithms have been developed to obtain unconstrained maximum-likelihood estimates of G, but none of these algorithms lead to estimates when functional constraints are present. In many cases, there is a natural interest in functional ?(G), such as the mean and variance, of the mixing distribution, and profile likelihoods and confidence intervals for ?(G) are desired. In this paper we develop a penalized generalization of the ISDM algorithm of Kalbfleisch and Lesperance (1992) that can be used to solve the problem of constrained estimation. We also discuss its use in various different applications. Convergence results and numerical examples are given for the generalized ISDM algorithm, and asymptotic results are developed for the likelihood-ratio test statistics in the multinomial case.     

A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let X_i≤?≤X_m and Y_i≤?≤Y_n be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let R_i denote the rank of X_i in the combined order sample. Steck (1980) has found an expression for P(b_i≤R_i≤a_i, all i) when F = h(G), h being the incomplete beta function with parameters (α,β?α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.     

Comparison of statistical graph-size estimators

We investigate several unbiased estimators of the size of a graph G which are based on sampled induced subgraphs, stars and dyads. By comparing their variances we find some general dominance relations between them, and we also give some conditions on G which guarantee some other instances of dominance.