球对称分布一些子类的特征函数

本文用统一的方法给出了对称Kotz型分布,多元PearsonⅡ分布和椭球logistic对称分布.我们的结果是Iyenger and Tong的结果的推广,另外我们得到了PearsonⅡ型的特征函数的另一种形式.同时,我们讨论了相关的问题.     

Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.     

On the theory of elliptically contoured distributions

The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality.     

Singular random matrix decompositions: distributions

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.     

Variable orderings and the size of OBDDs for random partially symmetric Boolean functions

The size of ordered binary decision diagrams (OBDDs) strongly depends on the chosen variable ordering. It is an obvious heuristic to use symmetric variable orderings, i.e., variable orderings where symmetric variables are arranged adjacently. In order to evaluate this heuristic, methods for estimating the OBDD size for random partially symmetric functions are presented. Characterizations of cases where, with high probability, only symmetric variable orderings and, with high probability, only nonsymmetric variable orderings lead to minimum OBDD size are obtained. For this analysis estimates for the number of different blocks of random Boolean matrices are used. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 49–70, 1998     

Testing for affine equivalence of elliptically symmetric distributions

Let X and Y be d-dimensional random vectors having elliptically symmetric distributions. Call X and Y affinely equivalent if Y has the same distribution as AX+b for some nonsingular d×d-matrix A and some . This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data.     

A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property

Methodology and Computing in Applied Probability - We introduce a new class of multivariate elliptically symmetric distributions including elliptically symmetric logistic distributions and Kotz...     

从偏态Pearson Ⅶ分布生成的新的多元偏态t分布

一般而言,偏态的椭球等高分布是一类分布族,有相当一部分的分布都是积分形式,且此类积分不易求出,而偏态的正态、偏态的正态尺度混合、偏态的PⅦ型、偏态的PⅡ型的分布却有着很好的结构,偏态t分布属于偏态PⅦ型分布,因此,本文在偏态PⅦ型分布的基础上着重研究新的偏态t分布,给出它的背景、定义、两种随机表示及其等价性.     

Perfect elimination orderings for symmetric matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.

     

Data Sphering: Some Properties and Applications

Centring-then-sphering is a very important pretreatment in data analysis. The purpose of this paper is to study the asymptotic behavior of the sphering matrix based on the square root decomposition (SRD for short) and its applications. A sufficient condition is given under which SRD has nondegenerate asymptotic distribution. As examples, some commonly used and affine equivariant estimates of the dispersion matrix are shown to satisfy this condition. The case when the population dispersion matrix varies is also treated. Applications to projection pursuit (PP) are presented. It is shown that for elliptically symmetric distributions the PP index after centring-then-sphering is independent of the underlying population location and dispersion.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

On the convergence of the cyclic Jacobi method

In this paper we develop a class of cyclic orderings and prove that the cyclic Jacobi method using any cyclic ordering from this class converges to the eigensystem of a symmetric matrix.     

An application of Kendall distributions and alternative dependence measures: SPX vs. VIX

Most of the recently-defined notions of positive or negative dependence rely upon a variety of orderings of bivariate random vectors. These orderings are generally partial orders, and thus there are many pairs of random vectors which are not comparable. By using a weakened version of stochastic domination and the concepts of Kendall distributions and metacopulas, an entirely new class of orderings, in which the comparability issue is resolved, has been recently created. Each ordering in this class can be used to construct a measure of dependence. A detailed example will be given, using data from the Standard & Poor’s 500 index and Chicago Board of Trades index for implied volatility.     

Ordered affine Hjelmslev planes

Preorderings and orderings of affine Hjelmslev planes were defined in [3]. The ordering of an AH-plane is shown to induce an ordering on each coordinate biternary ring which is compatible with the multiplication of the associated algebra. Even stronger projective order relations, on both the planes and biternary rings, are introduced and are shown to be also compatible with the ternary operations.The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada, and the invaluable assistance of Dr. N. D. Lane, McMaster University.     

Extension of orderings on -fields

An analysis is made of the ways in which a total ordering of the set of symmetric elements of a skew field with involution can be extended to an ordering of a larger set of elements. This is done for several different types of orderings found in the literature.

     

从SPVⅡ分布生成的新的多元偏态t分布的有关性质

随机向量的t分布属于椭球等高分布族,然而,它是对称分布.在许多诸如经济学、生理学、社会学等领域中,有时回归模型中的随机误差不再满足对称性,通常表现出高度的偏态性(skewness).于是就有了偏态椭球等高分布族.本文在已有的多元偏态t分布的基础上,着重研究它的分布性质,包括线性组合分布、边缘分布、条件分布及各阶矩.     

Ordering of risks: a review

In this review we will consider and discuss the most important partial orderings of riks, namely consistent partial ordering and net-stop-loss ordering. More especially we will study the consequencies of ordering of risks for the compound risk: S = X₁+X₂+···+X_N.The impact of orderings of claim size distributions (F_X) and claim intensities (F_N) on orderings of claim amounts (F_S) is examined. The consequencies of these kind of orderings on orderings of risks by means of premium calculation principles is also discussed.In this framework the influence of the dangerousness of distributions on orderings of risks is given.In analogy with the notion of stochastic dominance appearing in the theory of finance, the notion of stop-loss dominance is introduced.     

多项分布总体的相等对增凸序问题的多样本检验

概率分布间随机序在实践中已经得到了广泛的应用,而且似然比检验是用以检验涉及随机序问题的最普遍的检验方法．但是,关于多个多项式总体间的增凸序约束的统计推断问题并没有得到充分发展．多样本的增凸序对无约束的检验问题已被研究．然而,多总体的相等性对增凸序的假设检验问题似乎更有研究意义．并且分布的相等对随机序的假设检验问题往往是统计学家最为普遍地考虑．对多样本的情况,本文考虑了分布的相等对增凸序的假设检验问题,并且获得似然比检验统计量的零渐近分布,它是一组x~2分布随机变量的加权和,即■~2分布．     

Mean–variance–skewness efficient surfaces,Stein’s lemma and the multivariate extended skew-Student distribution

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.     

On Pearson-Kotz Dirichlet distributions

In this paper, we discuss some basic distributional and asymptotic properties of the Pearson-Kotz Dirichlet multivariate distributions. These distributions, which appear as the limit of conditional Dirichlet random vectors, possess many appealing properties and are interesting from theoretical as well as applied points of view. We illustrate an application concerning the approximation of the joint conditional excess distribution of elliptically symmetric random vectors.