Super-simple cyclic designs with small values

Super-simple cyclic designs are useful on constructing codes and designs such as superimposed codes, perfect hash families and optical orthogonal codes with index two. In this paper, we show that there exists a super-simple cyclic (v,4,λ)$(v,4,\lambda )$ for 7?v?41$7?v?41$ and all admissible λ$\lambda$ with two definite exceptions of (v,λ)=(9,3),(13,5)$(v,\lambda )=(9,3),(13,5)$ and one possible exception of (v,λ)=(39,18)$(v,\lambda )=(39,18)$. Some useful algorithms are explained for computer search and new designs are displayed.     

Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA(N,k,s,t)$\mathrm{OA}(N,k,s,t)$'s is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of the system of equations. A branch-and-cut algorithm developed by Margot [2002. Pruning by isomorphism in branch-and-cut. Math. Programming Ser. A 94, 71–90; 2003a. Exploiting orbits in symmetric ILP. Math. Programming Ser. B 98, 3–21; 2003b. Small covering designs by branch-and-cut. Math. Programming Ser. B 94, 207–220; 2007. Symmetric ILP: coloring and small integers. Discrete Optim., 4, 40–62] for solving integer programming problems with large symmetry groups is used to find all non-isomorphic OA(24,7,2,2)$\mathrm{OA}(24,7,2,2)$'s, OA(24,k,2,3)$\mathrm{OA}(24,k,2,3)$'s for 6?k?11$6?k?11$, OA(32,k,2,3)$\mathrm{OA}(32,k,2,3)$'s for 6?k?11$6?k?11$, OA(40,k,2,3)$\mathrm{OA}(40,k,2,3)$'s for 6?k?10$6?k?10$, OA(48,k,2,3)$\mathrm{OA}(48,k,2,3)$'s for 6?k?8$6?k?8$, OA(56,k,2,3)$\mathrm{OA}(56,k,2,3)$'s, OA(80,k,2,4)$\mathrm{OA}(80,k,2,4)$'s, OA(112,k,2,4)$\mathrm{OA}(112,k,2,4)$'s, for k=6,7$k=6,7$, OA(64,k,2,4)$\mathrm{OA}(64,k,2,4)$'s, OA(96,k,2,4)$\mathrm{OA}(96,k,2,4)$'s for k=7,8$k=7,8$, and OA(144,k,2,4)$\mathrm{OA}(144,k,2,4)$'s for k=8,9$k=8,9$. Further applications to classifying covering arrays with the minimum number of runs and packing arrays with the maximum number of runs are presented.     

Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g  -matrix. This paper deals with the construction of D-optimal g(n,1)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

Skew Dyck paths,area, and superdiagonal bargraphs

Skew Dyck paths are a generalization of ordinary Dyck paths, defined as paths using up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, starting and ending on the x-axis, never going below it, and so that up and left steps never overlap. In this paper we study the class of these paths according to their area, extending several results holding for Dyck paths. Then we study the class of superdiagonal bargraphs, which can be naturally defined starting from skew Dyck paths.     

Skew Dyck paths

In this paper we study the class S$\mathbb{S}$ of skew Dyck paths, i.e. of those lattice paths that are in the first quadrant, begin at the origin, end on the x-axis, consist of up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, and such that up steps never overlap with left steps. In particular, we show that these paths are equinumerous with several other combinatorial objects, we describe some involutions on this class, and finally we consider several statistics on S$\mathbb{S}$.     

Cyclic quasi-symmetric designs and self-orthogonal codes of length 63

The enumeration of binary cyclic self-orthogonal codes of length 63 is used to prove that any cyclic quasi-symmetric 2-(63,15,35)$2\text{-}(63,15,35)$ design with block intersection numbers x=3$x=3$ and y=7$y=7$ is isomorphic to the geometric design having as blocks the three-dimensional subspaces in PG(5,2)$\text{<italic>PG</italic>}(5,2)$.     

Estimation of the Sobol indices in a linear functional multidimensional model

We consider a functional linear model where the explicative variables are known stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L²([0,1])${\mathbb{L}}^2([0,1])$. We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U-statistics. We prove the asymptotic normality and the efficiency of our estimators and we compare them from a theoretical and practical point of view with classical estimators of Sobol indices.     

Combinatorially composing Chebyshev polynomials

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n?0$m,n?0$, _Tm(_Tn(x))=_Tmn(x)$T_m(T_n(x))=T_{\mathit{mn}}(x)$ and _Um−1(_Tn(x))_Un−1(x)=_Umn−1(x).$U_{m-1}(T_n(x))U_{n-1}(x)=U_{\mathit{mn}-1}(x).$     

Laplace record data

Basic properties of upper record values _XT(1),_XT(2),…,_XT(n)$X_{T(1)},X_{T(2)},\dots ,X_{T(n)}$ from a symmetric two-parameter Laplace distribution are established. In particular, unimodality of the density function and the exact expression of the mode are derived. Moreover, we obtain approximations of the first and second moment and the variance of _XT(k)$X_{T(k)}$ which provide close approximations even for moderate k. Additionally, limit laws and simulation of Laplace records are considered. Finally, we discuss maximum likelihood estimation in a location-scale family of Laplace distributions. We obtain nice representations of the estimators provided that the location parameter is unknown and present interesting properties of the established estimators. Some illustrative examples complete the presentation.     

On E(s)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E(s²)$E(s^2)$ criterion. In this paper, improved lower bounds on E(s²)$E(s^2)$ are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E(s²)$E(s^2)$-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E(s²)$E(s^2)$-optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E(s²)$E(s^2)$-optimal designs.     

Efficiency in estimation of memory

We study the efficiency of semiparametric estimates of memory parameter. We propose a class of shift invariant tapers of order (p,q). For a fixed p, the variance inflation factor of the new tapers approaches 1 as q   goes to infinity. We show that for d∈(−1/2,p+1/2)$d\in (-1/2,p+1/2)$, the proposed tapered Gaussian semiparametric estimator has the same limiting distribution as the nontapered version for d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The new estimator is mean and polynomial trend invariant, and is computationally advantageous in comparison to the recently proposed exact local Whittle estimator. The simulation study shows that our estimator has comparable or better mean squared error in finite samples for a variety of models.     

Testing for random effects and serial correlation in spatial autoregressive models

This paper constructs and evaluates tests for random effects and serial correlation in spatial autoregressive panel data models. In these models, ignoring the presence of random effects not only produces misleading inference but inconsistent estimation of the regression coefficients. Two different estimation methods are considered: maximum likelihood and instrumental variables. For each estimator, optimal tests are constructed: Lagrange multiplier in the first case; Neyman's C(α)$C(\alpha )$ in the second. In addition, locally size-robust tests, for individual hypotheses under local misspecification of the unconsidered parameter, are constructed. Extensive Monte Carlo evidence is presented.     

Local asymptotic minimax theory for block-decreasing densities

In this paper, we study Lebesgue densities on (0,∞)^d$(0,\infty {)}^d$ that are non-increasing in each coordinate, while keeping all other coordinates fixed, from the perspective of local asymptotic minimax lower bound theory. In particular, we establish a local optimal rate of convergence of the order n^−1/(d+2)$n^{-1/(d+2)}$.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.