The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by I_R^*[r] the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set I_R^*[r] for any positive integer r≡1 (mod 3) (only nine cases are left undecided for r=7,13,16,19), and establish that I_R^*[r]=J[r] for r≡1 (mod 3) and r22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

On Berry-Esseen theorem for some functions of U-statistics

The rate of convergence in the central limit theorem and in the random central limit theorem for some functions of U-statistics are established. The theorems refer to the asymptotic behaviour of the sequence {g(U_n),n≥1}, where g belongs to the class of all differentiable functions g such that g′εL(δ) and U_n is a U-statistics.     

Predicting integrals of diffusion processes

Consider predicting the integral of a diffusion process Z in a bounded interval A, based on the observations Z(t_1n),…,Z(t_nn), where t_1n,…,t_nn is a dense triangular array of points (the step of discretization tends to zero as n increases) in the bounded interval. The best linear predictor is generally not asymptotically optimal. Instead, we predict using the conditional expectation of the integral of the diffusion process, the optimal predictor in terms of minimizing the mean squared error, given the observed values of the process. We obtain that, conditioning on the observed values, the order of convergence in probability to zero of the mean squared prediction error is O_p(n⁻²). We prove that the standardized conditional prediction error is approximately Gaussian with mean zero and unit variance, even though the underlying diffusion is generally non-Gaussian. Because the optimal predictor is hard to calculate exactly for most diffusions, we present an easily computed approximation that is asymptotically optimal. This approximation is a function of the diffusion coefficient.     

The selection of zero-non-zero patterned cointegrating vectors in error-correction modelling

An effective and efficient search algorithm has been developed to select from an 1(1) system zero-non-zero patterned cointegrating and loading vectors in a subset VECM, B^q(l)y(t-1) + B^q-1 (L)Ay(t) = ε( t ) , where the long term impact matrix Bq(l) contains zero entries. The algorithm can be applied to higher order integrated systems. The Finnish money-output model presented by Johansen and Juselius (1990) and the United States balanced growth model presented by King, Plosser, Stock and Watson (1991) are used to demonstrate the usefulness of this algorithm in examining the cointegrating relationships in vector time series.     

Adaptive wavelet empirical Bayes estimation of a location or a scale parameter

Assume that in independent two-dimensional random vectors (X₁,θ₁),…,(X_n,θ_n), each θ_i is distributed according to some unknown prior density function g. Also, given θ_i=θ, X_i has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ⁻¹q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (X_n+1,θ_n+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth.     

Likelihood ratio ordering of order statistics

This paper provides an improvement on the work of Bapat and Kochar (1994, Linear Algebra Appl., 199, 281–291) and strengthens the literature on the likelihood ratio ordering of order statistics. For independent (but possibly nonidentically distributed) absolutely continuous random variables X₁,…,X_n, it is shown under some weak conditions that

X_1:n^lrlrX_n:n,

where ^lr stands for the likelihood ratio ordering and X_k:n represents the kth-order statistic.     

Linearity of regression for non-adjacent record values

Let X₁,X₂,… be a sequence of iid random variables having a continuous distribution; by R₁,R₂,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(R_m+k|R_m) or E(R_m|R_m+k) are identified.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).     

A critical look at Lo's modified R/S statistic

We report on an empirical investigation of the modified rescaled adjusted range or R/S statistic that was proposed by Lo, 1991. Econometrica 59, 1279–1313, as a test for long-range dependence with good robustness properties under ‘extra’ short-range dependence. In contrast to the classical R/S statistic that uses the standard deviation S to normalize the rescaled range R, Lo's modified R/S-statistic V_q is normalized by a modified standard deviation S_q which takes into account the covariances of the first q lags, so as to discount the influence of the short-range dependence structure that might be present in the data. Depending on the value of the resulting test-statistic V_q, the null hypothesis of no long-range dependence is either rejected or accepted. By performing Monte-Carlo simulations with ‘truly’ long-range- and short-range dependent time series, we study the behavior of V_q, as a function of q, and uncover a number of serious drawbacks to using Lo's method in practice. For example, we show that as the truncation lag q increases, the test statistic V_q has a strong bias toward accepting the null hypothesis (i.e., no long-range dependence), even in ideal situations of ‘purely’ long-range dependent data.     

On the non-existence of generalised Hadamard matrices

The determinant of a generalized Hadamard matrix over its group ring factored out by the relation Σ_gεG G = 0 is shown to have certain number theoretic properties. These are exploited to prove the non-existence of many generalised Hadamard matrices for groups whose orders are divisible by 3, 5 or 7. For example the GH(15, C₁₅), GH(15, C₃) and GH(15, C₅) do not exist. Also for certain n and G we find the set of determinants of the GH(n, G) matrices.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

Some new partial orderings on Sn and Sn×Sn

A new class of partial orderings on S_n, the set of permutations of {1,…,n}, is given. Each of these partial orderings is shown to be a subordering of a recently described partial ordering on S_n (Block, Chhetry, Fang and Sampson (1990)) which is related to Schriever's (1987) more associated ordering on bivariate distributions. Also given is an extension of three known partial orders on S_n to partial orders on S_n×S_n. These extensions facilitate the study of functions from S_n×S_n into , which preser these partial orderings, thereby, providing a methodology for extending the notion of arrangement increasing functions.     

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2^m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.     

Regular group divisible designs and Bhaskar Rao designs with block size three

Some recursive constructions are given for Bhaskar Rao designs. Using examples of these designs found by Shyam J. Singh, Rakesh Vyas and new ones given here we show the necessary conditions λ≡0 (mod 2), λυ(υ?1)≡0 (mod 24) are sufficient for the existence of Bhaskar Rao designs with one association class and block size 3. This result is used with a result of Street and Rodger to obtain regular partially balanced block designs with 2υ treatments, block size 3, λ₁=0, group size 2 and υ groups.     

NECESSARY AND SUFFICIENT CONDITION FOR A BALANCED ARRAY OF STRENGTH 21 TO BE A BALANCED FRACTIONAL 2 FACTORIAL DESIGN OF RESOLUTION 2l

A necessary and sufficient condition for a balanced array of strength 2l to be a balanced fractional 2^m factorial design of resolution 2l is given. This design has the property that the main effects, two-factor interactions,.and (l-1)-factor interactions are estimable ignoring the (l + 1)-factor and higher order interactions, and that the covariance matrix of their estimates is invariant under any permutation of m factors. The above condition includes sufficient conditions given in earlier works of Shirakura (1976b, 1977).     

Asymptotic normality of a weighted integrated squared error of kernel regression estimates with data-dependent bandwidth

Let (X, Y) be a bivariate random vector and let be the regression function of Y on X that has to be estimated from a sample of i.i.d. random vectors (X₁, Y₁),…,(X_n, Y_n) having the same distribution as (X, Y). In the present paper it is shown that the normalized integrated squared error of a kernel estimator with data-driven bandwidth is asymptotically normally distributed.