关于完全t部图的色等价性

设K（n1,n2,…,nt）表示完全t部图,K(n1,n2,…,nt）－A表示从K（n1,n2,…,nt）中删去子边集A所得之图．本文证明了：令G＝K（n1,n2,…,nt）,J为整数集,R为实数集．设简单图Y满足Y～G,则且进一步有：若s＞0且αi∈R（i＝1,2…．t）．则     

Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n ? 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.

The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.     

Fitting models to data: Interaction versus polynomial? your choice

Polynomials are widely used for fitting models empirically to data. Low-degree polynomials (specifically, degrees 1, 2, and at most 3) have stood the test of time by proving their versatility when it comes to fitting a wide variety of different surface shapes over limited regions of interest. However, when faced with modeling a surface over an experimental region whose boundaries extend beyond some localized neighborhood or limited-sized region of interest, a polynomial of degree 2, or even of degree 3, may not be adequate. For this situation we propose fitting an interaction model which is a reduced form of higher-degree polynomial. Some examples of actual experiments are presented to illustrate the improvement in fit by an interaction model over that of a standard polynomial, even for response surfaces with uncomplicated shapes.     

Minimum Cost Edge Subset Covering Exactly k Vertices of a Graph

Given a graph G with nonnegative edge costs and an integer k, we consider the problem of finding an edge subset S of minimum total cost with respect to the constraint that S covers exactly k vertices of G. An O(n ³) algorithm is presented where n is the order of G. It is based on the author's previous paper dealing with a similar problem asking S to cover at least k vertices.     

A-optimal designs for mixture central polynomial model with qualitative factors

Mixture central polynomial models with qualitative factors are widely applied in many fields of research. In this paper, a method of finding A-optimal design for two degree mixture central polynomial model with qualitative factors will be proposed. The variance function will be given for getting the support points of the design. The A-optimality is confirmed by the equivalence theorem. In addition, this method also works effectively with higher degree models.     

On the distribution of a second degree polynomial statistic in singular normal var1aies

This note provides an alternative proof of a basic theorem on the distribution of a second degree polynomial statistic in noncentral singular normal variates originally proven by Rayner and Livingstone (1965)     

Recurrence relations of coefficients of the generalized hypergeometric function in multivariate analysis

James(1960) defined the zonal polynomials and used it to represent the joint distributions of latent roots of VVisfiart matrix. The zonal polviionnals played an important role to define the generalized hypergeometric function of symmetric matrix argument Since then, many density functions and moments based on Wishart matrix have been expressed in terms of the generalized hy¬pergeometric Function. The purpose of this paper is to get the recurrence relations for the coefficients of it. In Section 1 we derive a partial differen¬tial equations having the generalized hypergeometric function as the unique solution. Then we ubtain the recurrence relations until order 7 in Section 2.     

A simple bootstrap bandwidth selector for local polynomial fitting

A new, fully data-driven bandwidth selector with a double smoothing (DS) bias term and a data-driven variance estimator is developed following the bootstrap idea. The data-driven variance estimation does not involve any additional bandwidth selection. The proposed bandwidth selector convergences faster than a plug-in one due to the DS bias estimate, whereas the data-driven variance improves its finite sample performance clearly and makes it stable. Asymptotic results of the proposals are obtained. A comparative simulation study was done to show the overall gains and the gains obtained by improving either the bias term or the variance estimate, respectively. It is shown that the use of a good variance estimator is more important when the sample size is relatively small.     

On location and scale parameters of exponential distributions with censored observations

In this paper we first address the problem of estimating the common scale of several exponential distributions with unknown location parameters when censored samples are observed. The improved estimators are basically Stein type testimators. These testimators are then used to construct improved estimators of location parameters.     

Estimation of parameters of a polynomial model under intra class correlation structure for incomplete longitudinal data

The problem of analyzing and modeling incomplete longitudinal data arising from clinical and epidemiological studies are discussed, A method for handling arbitrarily missing observations under the intra class correlation structure and a polynomial model is developed. Explicit expressions for likelihood equations and information matrix for a second degree polynomial model are provided. The method is illustrated through an example.