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.     

Self-dual codes which are principal ideals of the group algebra F2[{F2m, + }]

This paper follows up Camion's contribution on self-dual codes which are principal ideals of the algebra F₂[{F_2^m, + }], the so-called H-codes. Our main result is that this class of codes does not meet the Gilbert-Varshamov bound. We obtain this result by giving an upper bound on the minimal distance of any H-code. We characterize extremal H-codes and link up their generators with certain difference sets.     

A construction for generalized hadamard matrices

We prove that if p^r and p^r ? 1 are both prime powers then there is a generalized Hadamard matrix of order p^r(p^r ? 1) with elements from the elementary abelian group Z_p x?x Z_p. This result was motivated by results of Rajkundia on BIBD's. This result is then used to produce p^r ? 1 mutually orthogonal F-squares F(p^r(p^r ? 1); p^r ? 1).     

Nested Hadamard difference sets

A Hadamard difference set (HDS) has the parameters (4N², 2N² − N, N² − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z²_p^r contains a (hp^2r, √hp^r(2√hp^r − 1), √hp^r(√hp^r − 1)) HDS (HDS), p a prime not dividing |H| = h and p^j ≡ −1 (mod exp(H)) for some j, then H × Z²_p^t has a (hp^2t, √hp^t(2√hp^t − 1), √hp^t(√hp^t − 1)) HDS for every 0⩽t⩽r. Thus, if these families do not exist, we simply need to show that H × Z²_p does not support a HDS. We give two examples of families that are ruled out by this procedure.     

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.     

Optimal designs,supplementary difference sets and multipliers

We investigate multipliers of 2 - {v; q², q²; λ} supplementary difference sets where cyclotomy has been used to construct D-optimal designs.     

On approximating point spread distributions

We discuss some properties of the point spread distribution, defined as the distribution of the difference of two independent binomial random variables with the same parameter n including exact and approximate probabilities and related optimization issues. We use various approximation techniques for different distributions, special functions, and analytic, combinatorial and symbolic methods, such as multi-summation techniques. We prove that in case of unequal success rates, if these rates change with their difference kept fixed and small, and n is appropriately bounded, then the point spread distribution only slightly changes for small point differences. We also prove that for equal success rates p, the probability of a tie is minimized if p=1/2. Numerical examples are included for the case with n=12.     

New estimator for functions of the canonical correlation coefficients

Let ρ₁,…,ρ_p be the population canonical correlation coefficients from a normal distribution. This paper considers the estimation of δ₁,…,δ_p, where $\text{δ}{}_{\mathrm{i}}\text{=ρ}{}_{\mathrm{i}}{}^2\text{/(1−ρ}{}_{\mathrm{i}}{}^2\text{),}\text{i=1,…,p}$, in a decision theoretic way. Since the distribution of δ_i's is complicated, two-staged estimation has been a usual method so far; i.e., first find a good estimator of a matrix whose eigenvalues are the δ_i's, then use its eigenvalues as the estimators of δ_i's. In this paper we directly estimate δ_i's and evaluate the estimators with respect to a quadratic loss function. We propose a new class of estimators and prove its dominance over the usual estimator.     

Prediction intervals for growth curves

The growth curve model Y^n×p = A^{n×p ξ mtimes;k}B^k×p+ E^nxp, where Y is an observation matrix, &sigma is a matrix of unknown parameters, A is a known matrix of rank m, B is a known matrix of rank k with 1'= (1, …, 1) as its first row, and the rows of E are independent each distributed as N^p(0,Σ,) is considered. The problem of constructing the prediction intervals for future observations using the above model is considered and approximate intervals assuming different structures on σ are derived. The results are illustrated with several data sets.     

On block-transitive designs with affine automorphism group

Detailed necessary and sufficient conditions for a k-subset of AG(d, 3) to generate the block set of a block-transitive t-design with automorphism group AGL(d, 3) are derived for t = 3, 4, 5. Similar necessary conditions are found for the existence of a block-transitive design with automorphism group AGL(d, p) when p is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with k or v − k not exceeding 1000 were for t = 4 and (d, p) = (7, 3), (8, 3), and (3, 7). Examples of block-transitive 4-designs admitting AGL(7, 3) are found for each of the values k = 115, 116, 230, 437, and 552.     

Nonlinear orthogonal series estimates for random design regression

Let (X,Y) be a pair of random variables with supp(X)⊆[0,1] and EY²<∞. Let m be the corresponding regression function. Estimation of m from i.i.d. data is considered. The L₂ error with integration with respect to the design measure μ (i.e., the distribution of X) is used as an error criterion.Estimates are constructed by estimating the coefficients of an orthonormal expansion of the regression function. This orthonormal expansion is done with respect to a family of piecewise polynomials, which are orthonormal in L₂(μ_n), where μ_n denotes the empirical design measure.It is shown that the estimates are weakly and strongly consistent for every distribution of (X,Y). Furthermore, the estimates behave nearly as well as an ideal (but not applicable) estimate constructed by fitting a piecewise polynomial to the data, where the partition of the piecewise polynomial is chosen optimally for the underlying distribution. This implies e.g., that the estimates achieve up to a logarithmic factor the rate n^−2p/(2p+1), if the underlying regression function is piecewise p-smooth, although their definition depends neither on the smoothness nor on the location of the discontinuities of the regression function.     

Partitioned difference families and almost difference sets

Partitioned difference families (PDFs) were first studied by Ding and Yin in conjunction with the construction of constant composition codes (CCCs). In 2008, Yin et al. presented the constructions of a number of infinite classes of PDFs based on known difference sets in GF(q). In this paper, we further investigate the constructions of PDFs by using known almost difference sets in GF(q), and establish some recursive constructions of PDFs. As their applications, we also get a number of perfect difference systems of sets (DSSs) over Z_q² with q odd prime.     

A combined p-value test for multiple hypothesis testing

Tests that combine p-values, such as Fisher's product test, are popular to test the global null hypothesis H₀ that each of n component null hypotheses, H₁,…,H_n, is true versus the alternative that at least one of H₁,…,H_n is false, since they are more powerful than classical multiple tests such as the Bonferroni test and the Simes tests. Recent modifications of Fisher's product test, popular in the analysis of large scale genetic studies include the truncated product method (TPM) of Zaykin et al. (2002), the rank truncated product (RTP) test of Dudbridge and Koeleman (2003) and more recently, a permutation based test—the adaptive rank truncated product (ARTP) method of Yu et al. (2009). The TPM and RTP methods require users' specification of a truncation point. The ARTP method improves the performance of the RTP method by optimizing selection of the truncation point over a set of pre-specified candidate points. In this paper we extend the ARTP by proposing to use all the possible truncation points {1,…,n} as the candidate truncation points. Furthermore, we derive the theoretical probability distribution of the test statistic under the global null hypothesis H₀. Simulations are conducted to compare the performance of the proposed test with the Bonferroni test, the Simes test, the RTP test, and Fisher's product test. The simulation results show that the proposed test has higher power than the Bonferroni test and the Simes test, as well as the RTP method. It is also significantly more powerful than Fisher's product test when the number of truly false hypotheses is small relative to the total number of hypotheses, and has comparable power to Fisher's product test otherwise.     

Asymptotically optimal shrinkage estimates for non-normal data

Motivated by several practical issues, we consider the problem of estimating the mean of a p-variate population (not necessarily normal) with unknown finite covariance. A quadratic loss function is used. We give a number of estimators (for the mean) with their loss functions admitting expansions to the order of p ^?1/2 as p→∞. These estimators contain Stein's [Inadmissibility of the usual estimator for the mean of a multivariate normal population, in Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, Vol. 1, J. Neyman, ed., University of California Press, Berkeley, 1956, pp. 197–206] estimate as a particular case and also contain ‘multiple shrinkage’ estimates improving on Stein's estimate. Finally, we perform a simulation study to compare the different estimates.     

Redescending M-estimators

In finite sample studies redescending M-estimators outperform bounded M-estimators (see for example, Andrews et al. [1972. Robust Estimates of Location. Princeton University Press, Princeton]). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we study the optimality of redescending M-estimators. We show that redescending M-estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing over a class of densities the minimum variance sensitivity over a class of estimators. As a particular result, we prove that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides a guaranteed level of an estimator's variance sensitivity over the class of densities with a bounded variance.     

Robustness of locally most powerful invariant test for normal mixture model in control and treatment populations

Two independent samples from control with N(μ₁, σ²) and treatment with pN(μ₁, σ²) + (1 − p)N(μ₂, σ²) are considered. A locally most powerful invariant test for testing H₀: μ₁ = μ₂ against H₁ : μ₂ > μ₁, where σ² > 0, 0 < p < 1 are unknown, is obtained. Also, the robustness of the test statistic on the lines of Kariya and Sinha (Robustness of Statistical Tests (1989). Academic Press, New York) is studied.     

Efficient estimates in linear and nonlinear regression with heteroscedastic errors

In this paper we consider the heteroscedastic regression model defined by the structural relation Y = r(V, β) + s(W)ε, where V is a p-dimensional random vector, W is a q-dimensional random vector, β is an unknown vector in some open subset B of R^m, r is a known function from R^p × B into R, s is an unknown function on R^q, and ε is an unobservable random variable that is independent of the pair (V, W). We construct asymptotically efficient estimates of the regression parameter β under mild assumptions on the functions r and s and on the distributions of ε and (V, W).     

Poisson approximations in selected metrics by coupling and semigroup methods with applications

Let S_n = X₁ + … + X_n, where X₁,…, X_n are independent Bernoulli random variables. In this paper, we evaluate probability metrics of the Wasserstein type between the distribution of S_n and a Poisson distribution. Our results show that, if E(S_n) = O(1) and if the individual probabilities of success of the X_i's tend uniformly to zero, then the general rate of convergence of the above mentioned metrics to zero is O(∑ⁿ_i = ₁P²_i). We also show that this rate is sharp and discuss applications of these results.     

Exact two-sided Lieberman-Resnikoff sampling plans

We deal sith sampling by variables with two-way-protection in the case of aN(μσ²) distributed characteristic with unknown σ². For the sampling plan by Lieberman and Resnikoff (1955), which is based on the MVU estimator of the percent defective, we prove a formula for the OC. If the sampling parametersp ₁ (AQL),p ₂ (LQ) and α, β (type I, II errors) are given, we are able to compute the true type I and II errors of the usual (one-sided) approximation plans. Furthermore it is possible to compute exact two-sided Lieberman-Resnikoff sampling plans.