Finiteness of circulant weighing matrices of fixed weight

Let n be a fixed positive integer. Every circulant weighing matrix of weight n arises from what we call an irreducible orthogonal family of weight n. We show that the number of irreducible orthogonal families of weight n is finite and thus obtain a finite algorithm for classifying all circulant weighing matrices of weight n. We also show that, for every odd prime power q, there are at most finitely many proper circulant weighing matrices of weight q.     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.     

A comparison of lower bounds for the symmetric circulant traveling salesman problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on the optimal value that may be computed in polynomial time. We derive a new linear programming (LP) relaxation of the SCTSP from the semidefinite programming (SDP) relaxation in [E. de Klerk, D.V. Pasechnik, R. Sotirov, On semidefinite programming relaxation of the traveling salesman problem, SIAM Journal of Optimization 19 (4) (2008) 1559-1573]. Further, we discuss theoretical and empirical comparisons between this new bound and three well-known bounds from the literature, namely the Held-Karp bound [M. Held, R.M. Karp, The traveling salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138-1162], the 1-tree bound, and the closed-form bound for SCTSP proposed in [J.A.A. van der Veen, Solvable cases of TSP with various objective functions, Ph.D. Thesis, Groningen University, The Netherlands, 1992].     

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.     

Fractional normal inverse Gaussian diffusion

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.     

Only the first term of some series counts

Let X,X₁,X₂,… be i.i.d. random variables, and set S_n=X₁+?+X_n. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type ∑_n≥1a_nP(|S_n|≥xb_n) or ∑_n≥1a_nP(S_n≥xb_n) behave like their first term as x→∞.     

Empilrical likelihood for non-parametric regression models with missing responses: Multiple design case

Empirical likelihood(EL) ratio statistic on θ = g(x) is constructed based on the inverse probability weighted imputation approach in a nonparametric regression model Y = g(x) + ε(x ∈ [0,1]p) with fixed designs and missing responses,which asymptotically has χ12 distribution.This result is used to obtain a EL based confidence interval on θ.     

Numerical solution of random differential initial value problems: Multistep methods

This paper deals with the construction of numerical methods of random initial value problems. Random linear multistep methods are presented and sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a₁,a₂,...,a_n} and B = {b₁,b₂,...,b_m}, in which the numbers t_ai,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_ai,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_c,d,A, the number of vertices in A with degree between c and d of G_n,m∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_n,m∈ζ（n,m;{pk}） has maximum degree D（n）in A? under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A?