A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

Extending Mosaic Displays: Marginal,Conditional, and Partial Views of Categorical Data

Abstract

This article first illustrates the use of mosaic displays for the analysis of multiway contingency tables. We then introduce several extensions of mosaic displays designed to integrate graphical methods for categorical data with those used for quantitative data. The scatterplot matrix shows all pairwise (bivariate marginal) views of a set of variables in a coherent display. One analog for categorical data is a matrix of mosaic displays showing some aspect of the bivariate relation between all pairs of variables. The simplest case shows the bivariate marginal relation for each pair of variables. Another case shows the conditional relation between each pair, with all other variables partialled out. For quantitative data this represents (a) a visualization of the conditional independence relations studied by graphical models, and (b) a generalization of partial residual plots. The conditioning plot, or coplot shows a collection of partial views of several quantitative variables, conditioned by the values of one or more other variables. A direct analog of the coplot for categorical data is an array of mosaic plots of the dependence among two or more variables, stratified by the values of one or more given variables. Each such panel then shows the partial associations among the foreground variables; the collection of such plots shows how these associations change as the given variables vary.     

NA随机变量序列的Egorov型强大数律的等价条件

本文研究了NA随机变量的Egorov型强大数律.利用NA随机变量的概率不等式,得到了NA随机变量序列的Egorov型强大数律的一些等价条件,所获结果推广和改进了在独立随机变量序列的Egorov的结果和在NA随机变量序列已有的一些结果.     

An exact algorithm for solving large-scale two-stage stochastic mixed-integer problems: Some theoretical and experimental aspects

We present an algorithmic framework, so-called BFC-TSMIP, for solving two-stage stochastic mixed 0–1 problems. The constraints in the Deterministic Equivalent Model have 0–1 variables and continuous variables at any stage. The approach uses the Twin Node Family (TNF) concept within an adaptation of the algorithmic framework so-called Branch-and-Fix Coordination for satisfying the nonanticipativity constraints for the first stage 0–1 variables. Jointly we solve the mixed 0–1 submodels defined at each TNF integer set for satisfying the nonanticipativity constraints for the first stage continuous variables. In these submodels the only integer variables are the second stage 0–1 variables. A numerical example and some theoretical and computational results are presented to show the performance of the proposed approach.     

Tackling the omitted variables problem without the strong assumptions of proxies

Omitted variables that interact with included independent variables change the vertical placement of observations. Thus, by projecting the data to an output oriented VRS DEA frontier, the influence of omitted variables can be eliminated. After this is done once, the efficient observations can be eliminated and the process repeated. Each subsequent iteration shows the relationship between the dependant and known independent variable for progressively less favorable omitted variables. Building on these ideas, we introduce a new analytical technique named “Reiterative Truncated Projected Least Squares” (RTPLS). We provide both a theoretical argument and simulation evidence that RTPLS produces less bias than ordinary least squares (OLS) when there are omitted variables that interact with the included variables. By way of example, we show how omitted variables have affected the relationship between the monetary base (MB) and the money supply (M2 + CDs) for Japan using monthly data from January 1970 to April 2003.     

On the mean number of trials until the last trials satisfy a given condition

Random variables are collected one at a time until the last k variables satisfy a given condition. The mean waiting time until this happens is studied and a general lemma is given. If the condition is satisfied by many possible stopping sequences the probability distribution of the k last variables is discussed. A fairly general treatment is given for the case when k = 2. Two other special cases are mentioned, viz. the case when the condition is defined by order relations between the last variables, and the case when the variables are discrete.     

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.     

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

Choosing the best set of variables in regression analysis using integer programming

This paper is concerned with an algorithm for selecting the best set of s variables out of k(> s) candidate variables in a multiple linear regression model. We employ absolute deviation as the measure of deviation and solve the resulting optimization problem by using 0-1 integer programming methodologies. In addition, we will propose a heuristic algorithm to obtain a close to optimal set of variables in terms of squared deviation. Computational results show that this method is practical and reliable for determining the best set of variables.     

Worst-case global optimization of black-box functions through Kriging and relaxation

A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.     

Multiple Neutral Data Fitting

A method is proposed for estimating the relationship between a number of variables; this differs from regression where the emphasis is on predicting one of the variables. Regression assumes that only one of the variables has error or natural variability, whereas our technique does not make this assumption; instead, it treats all variables in the same way and produces models which are units invariant – this is important for ensuring physically meaningful relationships. It is thus superior to orthogonal regression in that it does not suffer from being scale-dependent. We show that the solution to the estimation problem is a unique and global optimum. For two variables the method has appeared under different names in various disciplines, with two Nobel laureates having published work on it.     

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

修正的SQP型并行变量分配算法

Ferris 和Mangasarian 提出求解最优化问题的PVD(并行变量分配)算法, 此算法是把变量分为主要变量和辅助变量, 分配到p个处理机上, 每个处理机除了负责更新本处理机的主要变量外, 同时还沿着给定的方向更新辅助变量, 使算法的鲁棒性和灵活性得到了很大的提高. 该文基于文献[6]提出一种修正的SQP型PVD算法, 构造其搜索方向是下降方向和可行方向的组合, 并对此方向给予一个高阶修正, 使此算法很好地防止 Maratos 效应发生, 而且能够克服在求解子问题时出现约束不相容的情况. 在合适的条件下, 推导出此算法具有全局收敛性.     

Using Bayesian networks for bankruptcy prediction: Some methodological issues

This study provides operational guidance for building naïve Bayes Bayesian network (BN) models for bankruptcy prediction. First, we suggest a heuristic method that guides the selection of bankruptcy predictors. Based on the correlations and partial correlations among variables, the method aims at eliminating redundant and less relevant variables. A naïve Bayes model is developed using the proposed heuristic method and is found to perform well based on a 10-fold validation analysis. The developed naïve Bayes model consists of eight first-order variables, six of which are continuous. We also provide guidance on building a cascaded model by selecting second-order variables to compensate for missing values of first-order variables. Second, we analyze whether the number of states into which the six continuous variables are discretized has an impact on the model’s performance. Our results show that the model’s performance is the best when the number of states for discretization is either two or three. Starting from four states, the performance starts to deteriorate, probably due to over-fitting. Finally, we experiment whether modeling continuous variables with continuous distributions instead of discretizing them can improve the model’s performance. Our finding suggests that this is not true. One possible reason is that continuous distributions tested by the study do not represent well the underlying distributions of empirical data. Finally, the results of this study could also be applicable to business decision-making contexts other than bankruptcy prediction.     

Substitutes and complements in network flow problems

If f is a function of several variables, one calls a pair of variables substitutes(complements) if the change of the value of the function when both variables are increased is at most (at least) equal to the sum of the changes when each is increased separately. We here consider the case where f is the value of a maximum weight circulation on a network and the variables are the upper and lower bounds and the weights of a pair of arcs. We introduce a simple combinatorial criterion for two arcs to be in “series” or “parallel” and show that these two cases correspond to the variables being complements or substitutes respectively. This generalizes results of Shapley for the special case of the maximum flow and optimal assignment problems. We also show that our result is best possible in that if two arcs are neither in series nor parallel, then the corresponding variables can be either substitutes or complements or both.     

关于两两NQD随机序列的一个极限定理

设为两两NQD随机序列, 且, 是一列严格单调递增的凸序列. 本文将 Feller (1946)关于独立同分布期望不存在随机序列的极限定理推广到两两NQD随机 序列的情形.     

The von Bahr–Esseen moment inequality for pairwise independent random variables and applications

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.     

Order statistics of uncertain random variables with application to <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.     

Isometry of functional spaces with different number of variables

We construct spaces of real functions of n + k variables that are isometric to spaces of real functions given on an n-dimensional Euclidean space. We present certain properties and examples of delta-like kernels used for the construction of isometric spaces of functions with different number of variables. We prove certain assertions that enable one to construct delta-like kernels with many variables by using delta-like kernels with smaller number of variables. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1027–1045, August, 1998.