On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

Regular automorphism groups on partial geometries

Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg(s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except (s, α) = (5, 2), and we have checked that it is true for s ⩽ 500.     

On a class of partially balanced incomplete block designs

Bose and Clatworthy (1955) showed that the parameters of a two-class balanced incomplete block design with λ1=1,λ2=0 and satisfying r <k can be expressed in terms of just three parameters r,k,t. Later Bose (1963) showed that such a design is a partial geometry (r,k,t). Bose, Shrikhande and Singhi (1976) have defined partial geometric designs (r,k,t,c), which reduce to partial geometries when c=0. In this note we prove that any two class partially balanced (PBIB) design with r <k, is a partial geometric design for suitably chosen r,k,t,c and express the parameters of the PBIB design in terms of r,k,t,c and λ2. We also show that such PBIB designs belong to the class of special partially balanced designs (SPBIB) studied by Bridges and Shrikhande (1974).     

Many (22, 44, 14, 7, 4) — and (15, 42, 14, 5, 4)-balanced incomplete block designs

A Balanced Incomplete Block Design (BIBD) is a pair (V, B) where V is a v-set and B is a collection of b k-subsets of V, called blocks, such that every element of V occurs in exactly r of the k-subsets and every 2-subset of V occurs in exactly λ of the blocks. The number of non-isomorphic designs of a BIBD (22, 44, 14, 7, 4) whose automorphism group is divisible by 7 or 11 are investigated. From this work, results are obtained on the number of non-isomorphic BIBDs (15, 42, 14, 5, 4).     

Orthogonal arrays and ordered designs

In the first part of this paper, we give a short and direct construction of signed orthogonal array SOA(λ,t,k,v), for any set of parameters λ,t,k,v,t ⩽ k. We also construct a specific basis of the Z-module generated by any SOA(0,t,k,v), for any t,k,v,t ⩽ k. We will then construct an inite family of large set of disjoint ordered designs by applying G.B. Khosrovshahi and S. Ajoodani-Namini's method to Luc. Teirlinck's construction.     

Arrays of strength s on two symbols

This paper gives necessary and sufficient conditions on σ, s, t and on μ, s, t for an array with s+t rows to have strength s and weight σ, or to be balanced and have strength s and weight μ. If a balanced array can exist, the conditions provide a construction. The solutions for t=1,2 are also given in an alternate form useful for the study of trim arrays. The balanced solution for t=1 is more detailed than that known so far, and permits one to determine whether or not a solution exists in possibly fewer steps.     

Maximum likelihood estimation for arma models in the presence of ARMA errors

An ARMA(p, q) process observed with an ARMA(c, d) error has an ARMA (p + c, k) representation with k = max(c + q, p + d) whose parameters satisfy some nonlinear constraints. Identification of the model is discussed. We develop Newton-Raphson estimators for the ARMA(p + c, k) process which take advantage of the information contained in the nonlinear restrictions. Explicit expressions for the derivatives of the restrictions are derived.     

Mixed covering arrays of strength three with few factors

Covering arrays with mixed alphabet sizes, or mixed covering arrays, are useful generalizations of covering arrays that are motivated by software and network testing. Suppose that there are k factors, and that the ith factor takes values or levels from a set G_i of size g_i. A run is an assignment of an admissible level to each factor. A mixed covering array, MCA(N;t,k,g₁g₂…g_k), is a collection of N runs such that for any t distinct factors, i₁,i₂,…,i_t, every t-tuple from G_i1×G_i2×?×G_it occurs in factors i₁,i₂,…,i_t in at least one of the N runs. When g=g₁=g₂=?=g_k, an MCA(N;t,k,g₁g₂…g_k) is a CA(N;t,k,g). The mixed covering array number, denoted by MCAN(t,k,g₁g₂…g_k), is the minimum N for which an MCA(N;t,k,g₁g₂…g_k) exists. In this paper, we focus on the constructions of mixed covering arrays of strength three. The numbers MCAN(3,k,g₁g₂…g_k) are determined for all cases with k∈{3,4} and for most cases with k∈{5,6}.     

An optimal k of kth MA-ARIMA models under a class of ARIMA model

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated ARIMA(p, d, q) model. We run a simulation using the three above examining methods under specific conditions. The main finding is that 5th exponential weighted moving average (5th EWMA) ARIMA model is the best forecasting model among others, which means the optimal k = 5. For Turkish Telecommunications (TTKOM) stock market, real data reveal the similar results of simulation study.     

Weight hierarchies of linear codes of dimension 3

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,…,d_k), where d_r is the smallest support of an r-dimensional subcode of C. The weight hierarchies of [n,3;q] codes are studied. In particular, for q⩽5 the possible weight hierarchies of [n,3;q] codes are determined.     

A construction of group divisible designs

Given any affine design with parameters v, b, r, k, λ and μ = k²/v and any design with parameters v′, b′, r′, k′, λ′ where r′ = tr for some natural number `t and k′?r, we construct a group divisible design with parameters v′' = vv′, m = v′, n = v, b′' = vb′, k′' = kk′, r′'= kr′, λ₁ = tkλ and λ₂ = μλ′. This is applied to some series of designs. As a lemma, we also show that any 0-1-matrix with row sums tr and column sums ?r may be written as the sum of r 0-1-matrices with row sums t and column sums ?1.     

Testing stability in a spatial unilateral autoregressive model

ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process X_{k, ?} = αX_{k ? 1, ?} + βX_{k, ? ? 1} + ?_{k, ?} is investigated and asymptotic normality with a scaling factor n^5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model X_k = α₁X_{k ? 1} + ??? + α_pX_{k ? p} + ?_k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α₁ + ??? + α_p is not normal.     

Defective Galton–Watson processes

The Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k ? 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑_{k ? 0}p_k = 1 ? ? for some ? ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ?. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ? → 0.     

Probabilities and moments of generalized discrete distributfons using finite difference operators

The probabilities and factorial moments of the univar iate and multivariate generalized (or compound) discrete di st r-Lbut Lons with probability generating functions H(t)=F(G(t)) and H(t₁,…,t_k)=F(G(t₁,…,t_k))or H(t₁,…,t_k) = F(G₁(t₁),…, G_k( t_k)) are derived using finite difference operators.     

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.     

Tabu search for covering arrays using permutation vectors

A covering array  CA(N;t,k,v)$\mathsf{CA}(N;t,k,v)$ is an N×k$N\times k$ array, in which in every N×t$N\times t$ subarray, each of the v^t$v^t$ possible t  -tuples over v$v$ symbols occurs at least once. The parameter t is the strength   of the array. Covering arrays have a wide range of applications for experimental screening designs, particularly for software interaction testing. A compact representation of certain covering arrays employs “permutation vectors” to encode v^t×1$v^t\times 1$ subarrays of the covering array so that a covering perfect hash family whose entries correspond to permutation vectors yields a covering array. We introduce a method for effective search for covering arrays of this type using tabu search. Using this technique, improved covering arrays of strength 3, 4 and 5 have been found, as well as the first arrays of strength 6 and 7 found by computational search.     

Some necessary conditions on the parameters of partial difference sets

In this paper, we study some necessary conditions on the parameters of nontrivial regular (υ, κ, λ, μ)-partial difference sets in abelian groups. In particular, we settle some undecided cases in Ma's table [Designs, Codes Cryptography, 4 (1994)]. Also, the case when λ ⩽ 1 is studied. Nonexistence results are obtained when λ = 0 and a complete characterization is given when λ = 1. Finally, parameters of partial difference sets with an odd μ are determined.     

Simulation of a stationary autoregression: A characterization of the normal distribution

Simulating a stationary AR(p), X_t = ∑^p_i=1α_iX_t−i + Z_t, when the innovations {Z_t} are assumed to be i.i.d. is straightforward. Starting the process in the stationary state, however, requires generation of (X₁,X₂,…,X_p) from the stationary p-dimensional distribution. When Z_t is normal this may be achieved by generating X_i as a linear function of X₁,X₂,…,X_i−1 and an independent normal variate for i = 2,3,…, p. It is shown that the ability to initialize a stationary AR(p) in this way characterizes the normal distribution.     

Testing multiple slippages

A class of invariant Bayes rules is derived for testing homogeneity of k (≥2) different populations against (^k_t) slippage alternatives that some (unknown) subset of size t of the given populations has parameter larger than the remaining k-t, where t is a given integer between 1 and k-1. For a similar problem in nonparametric situations, locally best tests based on ranks are derived.