An extension of the multivariate Chebyshev's inequality to a random vector with a singular covariance matrix

ABSTRACT

We extend Chebyshev's inequality to a random vector with a singular covariance matrix. Then we consider the case of a multivariate normal distribution for this generalization.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

回归模型中异性方差问题的克服

提出了回归分析中异性方差的检验和对估计结果带来的影响以及消除其异方差性的两种方法：对原模型进行变换和加权最小二乘法．     

Using mathematical programming to compute singlular multivariate normal probablities

This paper describes two new, mathematical programming-based approaches for evaluating general, one- and two-sidedp-variate normal probabilities where the variance-covariance matrix (of arbitrary structure) is singular with rankr(r<pand r and p can be of unlimited dimensions. In both cases, principal components are used to transform the original, ill-definedp-dimensional integral into a well-definedrdimensional integral over a convex polyhedron. The first algorithm that is presented uses linear programming coupled with a Gauss-Legendre quadrature scheme to compute this integral, while the second algorithm uses multi-parametric programming techniques in order to significantly reduce the number of optimization problems that need to be solved. The application of the algorithms is demonstrated and aspects of computational performance are discussed through a number of examples, ranging from a practical problem that arises in chemical engineering to larger, numerical examples.     

奇异点检测的小波方法在证券市场中的应用

本文使用小波变换模极大值方法分析次贷危机中美国证券市场的突变。研究发现,小波模极大值方法准确定位了金融资产价格异常点的具体时刻;检测出了2类奇异点,其中峰值点检测比过零点检测更稳健;这些奇异点对应了美国次贷危机主要发展阶段的重大经济事件,反应出危机中美国经济系统异常对金融市场造成的影响。文章最后进行了稳健性检验。     

和矩阵奇异值的极值表征

本文讨论两个m×n矩阵和的奇异值问题,给出了一个用矩阵迹的和表示奇异值之和的极值表达式。     

Comparison of Kernel Discriminant Coordinates Algorithms

The construction of kernel discriminant coordinates reduces to the solution of a generalized eigenvalue problem in which both matrices are nonnegative definite. Six different algorithms for solving that problem are described, and the performance of these algorithms is tested on 26 different datasets. The percentage of misclassifications using a linear discriminant function is noted, and the algorithms’ running times are ascertained. Classification is also performed in the space of classical discriminant coordinates.     

Choosing Principal Components: A New Graphical Method Based on Bayesian Model Selection

This article approaches the problem of selecting significant principal components from a Bayesian model selection perspective. The resulting Bayes rule provides a simple graphical technique that can be used instead of (or together with) the popular scree plot to determine the number of significant components to retain. We study the theoretical properties of the new method and show, by examples and simulation, that it provides more clear-cut answers than the scree plot in many interesting situations.     

The Joy of Copulas: Bivariate Distributions with Uniform Marginals

We describe a class of bivariate distributions whose marginals are uniform on the unit interval. Such distributions are often called “copulas.” The particular copulas we present are especially well suited for use in undergraduate mathematical statistics courses, as many of their basic properties can be derived using elementary calculus. In particular, we show how these copulas can be used to illustrate the existence of distributions with singular components and to give a geometric interpretation to Kendall's tau.