Multivariate comonotonicity

In this paper we consider several multivariate extensions of comonotonicity. We show that naive extensions do not enjoy some of the main properties of the univariate concept. In order to have these properties, more structures are needed than in the univariate case.     

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.     

Multivariate McCormick relaxations

McCormick (Math Prog 10(1):147–175, 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form \(F\circ f\) , where \(F\) is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions \(F\) , and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick’s result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed.     

Multivariate semi-Weibull distributions

Some multivariate semi-Weibull (denoted by MSW) distributions including the Marshall–Olkin multivariate semi-Weibull (denoted by MO-MSW) one are introduced. They are more general than the multivariate Weibull distributions proposed by Lee [L. Lee, Multivariate distributions having Weibull properties, J. Multivariate Anal. 9 (1979) 267–277]. The Marshall–Olkin multivariate semi-Pareto (denoted by MO-MSP) distribution is also defined. Two characterization theorems for the homogeneous MSW are proved. The multivariate minima domain of partial attraction of MSW is studied, and the interrelationships between MO-MSP and MSW are examined. The MSW distribution possesses the minima-semi-stability and minima-infinite divisibility properties.     

Multivariate CARMA processes

A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order (p,q$p,q$), q<p$q<p$, is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy process. Furthermore, various probabilistic properties of the state space model and the multivariate CARMA process itself are discussed in detail.     

Multivariate Lukacs theorem

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.     

多变量的SALZER定理

本文将在切触有理插值中起重要作用的Salzer定理推广到了多元的情形。     

Multivariate Lyapunov inequalities

We transfer here basic univariate Lyapunov inequalities to the multivariate setting of a shell by using the polar method.     

Multivariate Jackson Inequality

By known multivariate versions of the classical Jackson theorem, every compact cube P in R^N admits Jackson’s inequality. The purpose of this note is to deliver other examples of Jackson sets in R^N. We shall show that a finite union of disjoint Jackson compact sets in R^N is also a Jackson set and that this in general fails to hold for an infinite union of Jackson sets. We also give a characterization of Jackson sets in the family of Markov compact sets in R^N which together with a Bierstone result permits one to show that Whitney regular compact subsets of R^N are Jackson.     

基于正态分布的广义Liouville分布

对基于正态分布的广义Liouville分布族的性质和特征进行了研究,包括它的边缘分布,条件分布,特征函数和矩等,并对它们的正态性和独立性以及在回归分析中的应用展开了讨论