A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.     

关于零级超球级数的平均值

本文得出了零级超球级数的对数级和对数下级与其在椭圆上的平均值的关系式。     

De-aliasing effects using semifoldover techniques

Semifoldover designs, obtained by semifolding a regular two-level factorial design, have been discussed recently in the literature. In this article, with the use of indicator functions, we investigate various semifoldover designs that are obtained from a general two-level factorial design. We discuss when a main factor or a two-factor interaction can be de-aliased from their aliased two-factor interactions, and extend some of the existing results from regular designs to non-regular designs. Finally, we present some examples to illustrate the results developed here.     

Proof of Toft's Conjecture: Every Graph Containing No Fully Odd K4 is 3-Colorable

A fully odd K₄ is a subdivision of K₄ such that each of the six edges of the K₄ is subdivided into a path of odd length. In 1974, Toft conjectured that every graph containing no fully odd K₄ can be vertex-colored with three colors. The purpose of this paper is to prove Toft's conjecture.     

NONPARAMETRIC AUTOCOVARIANCE FUNCTION ESTIMATION

Nonparametric estimators of autocovariance functions for non-stationary time series are developed. The estimators are based on straightforward nonparametric mean function estimation ideas and allow use of any linear smoother (e.g. smoothing spline, local polynomial). The paper studies the properties of the estimators, and illustrates their usefulness through application to some meteorological and seismic time series.     

二次型与(半)稳定矩阵的判定

对二次型与（半）稳定矩阵之间的相互关系进行了讨论，得到了二者之间的相互判定方法，且利用二次型的不同配方法得到判定（半）稳定矩阵的一些条件     

Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

On Tiku's robust procedure — a Bayesian insight

Tiku's robust procedure for testing mean and variance from nonnormal universe is examined from the Bayesian viewpoint. The posterior distribution of the scale parameter is derived and then approximated by a Laguerre polynomial expansion while the posterior distribution of the location parameter is approximated by a linear combination of t-distributions. For the example with Darwin's data, the approximations appear to be extremely good.     

Convergence of Heavy-tailed Monte Carlo Markov Chain Algorithms

Abstract.  In this paper, we use recent results of Jarner & Roberts ( Ann. Appl. Probab., 12, 2002, 224) to show polynomial convergence rates of Monte Carlo Markov Chain algorithms with polynomial target distributions, in particular random-walk Metropolis algorithms, Langevin algorithms and independence samplers. We also use similar methodology to consider polynomial convergence of the Gibbs sampler on a constrained state space. The main result for the random-walk Metropolis algorithm is that heavy-tailed proposal distributions lead to higher rates of convergence and thus to qualitatively better algorithms as measured, for instance, by the existence of central limit theorems for higher moments. Thus, the paper gives for the first time a theoretical justification for the common belief that heavy-tailed proposal distributions improve convergence in the context of random-walk Metropolis algorithms. Similar results are shown to hold for Langevin algorithms and the independence sampler, while results for the mixing of Gibbs samplers on uniform distributions on constrained spaces are rather different in character.     

Characterizations based on Rényi entropy of order statistics and record values

Two different distributions may have equal Rényi entropy; thus a distribution cannot be identified by its Rényi entropy. In this paper, we explore properties of the Rényi entropy of order statistics. Several characterizations are established based on the Rényi entropy of order statistics and record values. These include characterizations of a distribution on the basis of the differences between Rényi entropies of sequences of order statistics and the parent distribution.