Noncentral quadratic forms of the skew elliptical variables

In this paper the quadratic forms in the skew elliptical variables are studied. A family of the noncentral generalized Dirichlet distributions is introduced and their distribution functions and probability density functions are obtained. The moment generating functions of the quadratic forms in the skew normal variables are obtained. Sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the noncentral generalized Dirichlet distributions are obtained. This leads to the noncentral Cochran's Theorem for the skew normal distribution.     

The Möbius distribution on the disc

A simple new family of distributions is proposed which has support the unit disc in two dimensions. The density functions of the family are unimodal, monotonic or uniantimodal. The bivariate symmetric beta distributions, which include the uniform distribution, are special cases, but many members of the family are skew. The distributions have three parameters, one controlling orientation, one controlling degree of concentration and the third controlling skewness, or more precisely off-centredness. Importantly, these parameters are globally orthogonal. An illustrative example of fitting the model to data is given. Conditional and marginal distributions are considered. The new distributions are compared favourably with an earlier suggestion of the same author.     

Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.     

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.     

斜球变量的F统计量

本文研究斜球变量的F统计量的性质并引进了一类斜F分布.我们得到了密度函数,分布函数,矩发生函数和矩,考察了F检验在斜正态分布族内的性质.     

The skew elliptical distributions and their quadratic forms

In this paper, a family of the skew elliptical distributions is defined and investigated. Some basic properties, such as stochastic representation, marginal and conditional distributions, distribution under linear transformations, moments and moment generating function are derived. The joint distribution of several quadratic forms is obtained. An example is given to show that the distributions of some statistics as the functions of the quadratic forms can be derived for various applications.     

Categorical Skew Lattices

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.     

Hook,line and sinker: A bijective proof of the skew shifted hook-length formula

A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall.Recently, the formula for skew straight shapes was proved by the author via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is a bijection that proves the whole family of hook-length formulas, and is also the simplest known bijective proof for shifted (non-skew) shapes. The complexity of the algorithm is studied, and a weighted generalization of Naruse’s formula is also presented.     

Skew normal measurement error models

In this paper we define a class of skew normal measurement error models, extending usual symmetric normal models in order to avoid data transformation. The likelihood function of the observed data is obtained, which can be maximized by using existing statistical software. Inference on the parameters of interest can be approached by using the observed information matrix, which can also be computed by using existing statistical software, such as the Ox program. Bayesian inference is also discussed for the family of asymmetric models in terms of invariance with respect to the symmetric normal distribution showing that early results obtained for the normal distribution also holds for the asymmetric family. Results of a simulation study and an analysis of a real data set analysis are provided.     

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.     

模糊一致体

定义了模糊一致体概念,研究了它与模糊拓扑体的关系及它与模糊一致空间的关系;给出了借助于体的模糊子集族对模糊一致体的刻画,还引入了模糊一致子体,模糊一致商体与模糊一致体的直积;并讨论了它们的分离性.     

The skew energy of a digraph

We are interested in the energy of the skew-adjacency matrix of a directed graph D, which is simply called the skew energy of D in this paper. Properties of the skew energy of D are studied. In particular, a sharp upper bound for the skew energy of D is derived in terms of the order of D and the maximum degree of its underlying undirected graph. An infinite family of digraphs attaining the maximum skew energy is constructed. Moreover, the skew energy of a directed tree is independent of its orientation, and interestingly it is equal to the energy of the underlying undirected tree. Skew energies of directed cycles under different orientations are also computed. Some open problems are presented.     

Analog of the cauchy function for a generalized equation with several deviations of the argument

We define a special multiplication of function series (skew multiplication) and a generalized Riemann-Stieltjes integral with function series as integration arguments. The generalized integrals and the skew multiplication are related by an integration by parts formula. The generalized integrals generate a family of linear generalized integral equations, which includes a family (represented in integral form via the Riemann-Stieltjes integral) of linear differential equations with several deviating arguments. A specific feature of these equations is that all deviating functions are defined on the same closed interval and map it into itself. This permits one to avoid specifying the initial functions and imposing any additional constraints on the deviating functions. We present a procedure for constructing the fundamental solution of a generalized integral equation. With respect to the skew multiplication, it is invertible and generates the product of the fundamental solution (a function of one variable) by its inverse function (a function of the second variable). Under certain conditions on the parameters of the equation, the product has all specific properties of the Cauchy function. We introduce the notion of adjoint generalized integral equation, obtain a representation of solutions of the original equation and the adjoint equation in generalized integral Cauchy form, and derive sufficient conditions for the convergence of solutions of a pair of adjoint equations.     

The generalized Cantor distribution and its corresponding inverse distribution

Building on ideas and concepts introduced by Lad, Taylor and Hosking, a generalized Cantor distribution and a corresponding skew generalized Cantor distribution are developed and analyzed. Associated inverse distributions are also introduced. In some cases method of moment estimation is shown to be readily implemented.     

A new construction for skew multivariate distributions

This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal (type A and type B) and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions.     

Skew Models I

If g and G are the pdf and the cdf of a distribution symmetric around 0 then the pdf 2g(u)G(λ u) is said to define a skew distribution. In this paper, we provide a mathematical treatment of the skew distributions when g and G are taken to come from one of Laplace, logistic, Student’s t, uniform, exponential power or the Bessel function distribution.      

对数收益率的偏斜Logistic分布与VaR估计

本文通过直方图和Q-Q图的直观方法展示了上证指数和深证指数的对数收益率具有尖峰厚尾和偏斜的分布特征,利用Shapiro-Wilk正态性检验和Kolmogorov-Smirnov检验等方法检验了对数收益率的分布与正态分布有显著性差异,并以较大的概率水平接受了对数收益率服从偏斜Logistic分布,同时给出了基于偏斜Logistic分布的VaR风险量的估计,结果显示上证指数的风险小于深证指数的风险。     

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.     

Probability density functions of some skew tent maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_a,b. We show that F_a,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known (Boyarsky A, Góra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Boston: Birkhauser, 1997), that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.     

尖峰厚尾保险损失数据的统计建模

保险损失数据的一个重要特点是尖峰厚尾性,即既有大量的小额损失,又有少量的高额损失,使得通常的损失分布模型很难拟合此类数据,从而出现了对各种损失分布模型进行改进的尝试.改进后的模型一方面要有较高的峰度,另一方面又要有较厚的尾部.最近几年文献中出现的改进模型主要是组合模型,即把一个具有非零众数的模型(如对数正态分布或威布尔分布)与一个厚尾分布模型(如帕累托分布或广义帕累托分布)进行组合.讨论了这些组合模型的性质和特点,并与偏t正态分布和偏t分布进行了比较分析,最后应用MCMC方法估计模型参数,并通过一个实际损失数据的拟合分析,表明偏t分布对尖峰厚尾损失数据的拟合要优于目前已经提出的各种组合模型.