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.     

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.     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

Linear codes from vectorial Boolean power functions

In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming codes and extended Hamming codes, respectively.     

Comparative study and sensitivity analysis of skewed spatial processes

Asymmetric spatial processes arise naturally in finance, economics, hydrology and ecology. For such processes, two different classes of models are considered in this paper. One of them, proposed by Majumdar and Paul (J Comput Graph Stat 25(3):727–747, 2016), is the Double Zero Expectile Normal (DZEXPN) process and the other is a version of the “skewed normal process”, proposed by Minozzo and Ferracuti (Chil J Stat 3:157–170, 2012), with closed skew normal multivariate marginal distributions. Both spatial models have useful properties in the sense that they are ergodic and stationary. As a brief treatise to test the sensitivity and flexibility of the new proposed DZEXPN model (Majumdar and Paul in J Comput Graph Stat 25(3):727–747, 2016), in relation to other skewed spatial processes in the literature using a Bayesian methodology, our results show that by adding measurement error to the DZEXPN model, a reasonably flexible model is obtained, which is also computationally tractable than many others mentioned in the literature. Meanwhile, we develop a full-fledged Bayesian methodology for the estimation and prediction of the skew normal process proposed in Minozzo and Ferracuti (Chil J Stat 3:157–170, 2012). Specifically, a hierarchical model is used to describe the skew normal process and a computationally efficient MCMC scheme is employed to obtain samples from the posterior distributions. Under a Bayesian paradigm, we compare the performances of the aforementioned three different spatial processes and study their sensitivity and robustness based on simulated examples. We further apply them to a skewed data set on maximum annual temperature obtained from weather stations in Louisiana and Texas.     

A New Robust Class of Skew Elliptical Distributions

Methodology and Computing in Applied Probability - A new robust class of multivariate skew distributions is introduced. Practical aspects such as parameter estimation method of the proposed class...     

Estimation of parameters in latent class models using fuzzy clustering algorithms

A mixture approach to clustering is an important technique in cluster analysis. A mixture of multivariate multinomial distributions is usually used to analyze categorical data with latent class model. The parameter estimation is an important step for a mixture distribution. Described here are four approaches to estimating the parameters of a mixture of multivariate multinomial distributions. The first approach is an extended maximum likelihood (ML) method. The second approach is based on the well-known expectation maximization (EM) algorithm. The third approach is the classification maximum likelihood (CML) algorithm. In this paper, we propose a new approach using the so-called fuzzy class model and then create the fuzzy classification maximum likelihood (FCML) approach for categorical data. The accuracy, robustness and effectiveness of these four types of algorithms for estimating the parameters of multivariate binomial mixtures are compared using real empirical data and samples drawn from the multivariate binomial mixtures of two classes. The results show that the proposed FCML algorithm presents better accuracy, robustness and effectiveness. Overall, the FCML algorithm has the superiority over the ML, EM and CML algorithms. Thus, we recommend FCML as another good tool for estimating the parameters of mixture multivariate multinomial models.     

Several classes of linear codes with few weights from the closed butterfly structure

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6], [29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19], we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction.     

Robust inference in linear mixed model with skew normal-symmetric error

Linear mixed effects models with general skew normal-symmetric (SNS) error are considered and several properties of the SNS distributions are obtained. Under the SNS settings, ANOVA-type estimates of variance components in the model are unbiased, the ANOVA-type F-tests are exact F-tests in SNS setting, and the exact confidence intervals for fixed effects are constructed. Also the power of ANOVA-type F-tests for components are free of the skewing function if the random effects normally distributed. For illustration of the main results, simulation studies on the robustness of the models are given by comparisons of multivariate skew-normal, multivariate skew normal-Laplace, multivariate skew normal-uniform, multivariate skew normal-symmetric, and multivariate normal distributed errors. A real example is provided for the illustration of the proposed method.     

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

ＳＯＭＥＭＵＬＴＩＶＡＲＩＡＴＥＤＭＲＬＡＮＤＮＢＵＥＤＥＦＩＮＩＴＩＯＮＳＢＡＳＥＤＯＮＣＯＮＤＩＴＩＯＮＡＬＳＴＯＣＨＡＳＴＩＣＯＲＤＥＲＷＡＮＧＹＵＥＤＯＮＧ（王跃东）ＣＡＯＪＩＮＨＵＡ（曹晋华）（ＤｅｐａｒｔｍｅｎｔｏｆＳｔａｔｉｓｔｉｃｓ，...     

反厄米特型Toeplitz线性方程组的反厄米特循环预处理子

本文主要研究了带位移的反厄米特型Toeplitz线性方程组Anx=b的一个新的反厄米特循环预处理子Cn,其中矩阵An的元素是函数f（θ）=a0＋ig（θ）的傅里叶系数．如果g（θ）是Wiener类实值函数,则矩阵Cn非奇异;且当n足够大时,矩阵（Cn^-1An）·（Cn^-1An）的谱以1为聚点,数值实验进一步显示了我们的预处理子是有效的．     

A multivariate skew normal distribution

In this paper, we define a new class of multivariate skew-normal distributions. Its properties are studied. In particular we derive its density, moment generating function, the first two moments and marginal and conditional distributions. We illustrate the contours of a bivariate density as well as conditional expectations. We also give an extension to construct a general multivariate skew normal distribution.     

Multivariate skew-symmetric distributions

In this paper, a class of multivariate skew distributions has been explored. Then its properties are derived. The relationship between the multivariate skew normal and the Wishart distribution is also studied.     

Factor models for multivariate count data

We develop a general class of factor-analytic models for the analysis of multivariate (truncated) count data. Dependencies in multivariate counts are of interest in many applications, but few approaches have been proposed for their analysis. Our model class allows for a variety of distributions of the factors in the exponential family. The proposed framework includes a large number of previously proposed factor and random effect models as special cases and leads to many new models that have not been considered so far. Whereas previously these models were proposed separately as different cases, our framework unifies these models and enables one to study them simultaneously. We estimate the Poisson factor models with the method of simulated maximum likelihood. A Monte-Carlo study investigates the performance of this approach in terms of estimation bias and precision. We illustrate the approach in an analysis of TV channels data.     

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.     

Linear codes using skew polynomials with automorphisms and derivations

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.     

A new test for multivariate normality

We propose a new class of rotation invariant and consistent goodness-of-fit tests for multivariate distributions based on Euclidean distance between sample elements. The proposed test applies to any multivariate distribution with finite second moments. In this article we apply the new method for testing multivariate normality when parameters are estimated. The resulting test is affine invariant and consistent against all fixed alternatives. A comparative Monte Carlo study suggests that our test is a powerful competitor to existing tests, and is very sensitive against heavy tailed alternatives.     

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.     

On Affine Equivariant Multivariate Quantiles

An extension of univariate quantiles in the multivariate set-up has been proposed and studied. The proposed approach is affine equivariant, and it is based on an adaptive transformation retransformation procedure. Behadur type linear representations of the proposed quantiles are established and consequently asymptotic distributions are also derived. As applications of these multivariate quantiles, we develop some affine equivariant quantile contour plots which can be used to study the geometry of the data cloud as well as the underlying probability distribution and to detect outliers. These quantiles can also be used to construct affine invariant versions of multivariate Q-Q plots which are useful in checking how well a given multivariate probability distribution fits the data and for comparing the distributions of two data sets. We illustrate these applications with some simulated and real data sets. We also indicate a way of extending the notion of univariate L-estimates and trimmed means in the multivariate set-up using these affine equivariant quantiles.     

Distribution of quadratic forms under skew normal settings

For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results.