Combining marginal probability distributions via minimization of weighted sum of Kullback-Leibler divergences

This paper deals with the problem of combining marginal probability distributions as a means for aggregating pieces of expert information. A novel approach, which takes the combining problem as an analogy of statistical estimation, is proposed and discussed. The combined distribution is then searched as a minimizer of a weighted sum of Kullback-Leibler divergences of the given marginal distributions and corresponding marginals of the searched one. Necessary and sufficient conditions for a distribution to be a minimizer are stated. For discrete random variables an iterative algorithm for approximate solution of the minimization problem is proposed and its convergence is proved.     

Jackknife variance estimators of the location estimator in the one-way random-effects model

In a one-way random-effects model, we frequently estimate the variance components by the analysis-of-variance method and then, assuming the estimated values are true values of the variance components, we estimate the population mean. The conventional variance estimator for the estimate of the mean has a bias. This bias can become severe in contaminated data. We can reduce the bias by using the delta method. However, it still suffers from a large bias. We develop a jackknife variance estimator which is robust with respect to data contamination.This research was supported by the Korea Science and Engineering Foundation.     

Uniform asymptotic normality of the matrix-variate Beta-distribution

With the upper bound of Kullback-Leibler distance between a matrix variate Beta-distribution and a normal distribution, this paper gives the conditions under which a matrix-variate Beta-distribution will approach uniformly and asymptotically a normal distribution.     

Applications of the growth characteristics induced by the spectral distance

Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C^?-algebra are equal if and only if they are quasinilpotent equivalent.     

On the degree distance of a graph

If G is a connected graph with vertex set V, then the degree distance of G, D^′(G), is defined as , where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound for graphs of order n and diameter d. As a corollary we obtain the bound for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].     

On the editing distance of graphs

An edge‐operation on a graph G is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs , the editing distance from G to is the smallest number of edge‐operations needed to modify G into a graph from . In this article, we fix a graph H and consider Forb(n, H), the set of all graphs on n vertices that have no induced copy of H. We provide bounds for the maximum over all n‐vertex graphs G of the editing distance from G to Forb(n, H), using an invariant we call the binary chromatic number of the graph H. We give asymptotically tight bounds for that distance when H is self‐complementary and exact results for several small graphs H. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:123–138, 2008     

On the distance function of a connected graph

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering

We show that the Euclidean Wasserstein distance between two compactly supported solutions of the one-dimensional porous medium equation having the same center of mass decays to zero for large times. As a consequence, we detect an improved -rate of convergence of solutions of the one-dimensional porous medium equation towards well-centered self-similar Barenblatt profiles, as time goes to infinity.

     

An Information-theoretic Approach to the Effective Usage of Auxiliary Information from Survey Data

In this paper, we propose an information-theoretic approach to the effective usage of auxiliary information from survey data, which is suitable for both simple and complex survey data. Our estimator under simple random sampling without replacement will be consistent and asymptotically normal. We show that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information. For more complex survey designs, the resulting estimator is in essence asymptotically equivalent to a pseudo empirical likelihood estimator. Results of a limited simulation study show that the proposed estimators perform well among a number of competitors.     

On the maps preserving the equality of distance

In this paper, in connection with the known result of Baker and Vogt, we get if preserves equality of distance with dimension X?2 and Y is strictly convex, the range of f contains a segment, then f is affine.     

Inverse of the distance matrix of a block graph

A connected graph G, whose 2-connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its distance matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D)?≠?0. We give a formula for both the determinant and the inverse, D ^?1 of D.     

The limiting normality of the test statistic for the two-sample problem induced by a convex sum distance

Critchlow (1992, J. Statist. Plann. Inference, 32, 325–346) proposed a method of a unified construction of a class of rank tests. In this paper, we introduce a convex sum distance and prove the limiting normality of the test statistics for the two-sample problem derived by his method.     

On measures of the distance of a mixture from its parent distribution

In a previous paper in this Journal, Heyde and Leslie [6] examined moment measures of the distance of a mixture from its parent distribution. They confined their attention to the case where the parent distribution is either normal or exponential, and related the moment measures to the more familiar uniform distance between distributions. In this paper we improve on their results by sharpening one of their inequalities. We then use new techniques to extend their investigation to a larger class of parent distributions.