Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions

A class of multivariate distributions that are mixtures of the positive powers of a max-infinitely divisible distribution are studied. A subclass has the property that all weighted minima or maxima belong to a given location or scale family. By choosing appropriate parametric families for the mixing distribution and the distribution being mixed, families of multivariate copulas with a flexible dependence structure and with closed form cumulative distribution functions are obtained. Some dependence properties of the class, as well as some characterizations, are given. Conditions for max-infinite divisibility of multivariate distributions are obtained.     

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACT

In this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.     

Maxima and minima of independent and non-identically distributed bivariate Gaussian triangular arrays

In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of ith vector of given nth row is the function of i/n. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.     

Three dimensional distribution of Brownian motion extrema

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

多维不可微平稳正态过程的极值

讨论了强相关多维不可微平稳正态过程的渐进性质 ,给出了其最大值的极限分布     

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.     

关于Gamma分布的秩序统计量的随机比较

对于独立不同分布的两个Gamma样本，当它们相同的形状参数大于或等于1时，最近Korwar(2002)证明了，当它们的尺度参数满足优化序时，样本的卷积就满足似然比序．本文我们证明了，当Gamma分布的形状参数小于1时，样本的对应秩序统计量之间存在一致的一般随机序；然而当形状参数大于1时，样本对应的极大值和极小值统计量有着相反的一般随机序。     

Location of hot spots in thin curved strips

The maxima and minima of Neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of Neumann eigenfunctions of the underlying curves. In particular, the hot spots conjecture for a new large class of domains (possibly non-convex and non-Euclidean) is proved.     

一类由概率分布生成的新度量

主要得到了一类由概率分布生成的新度量.以信息理论中的重要概念-相对熵为基础,对前人文章中的重要结论进行推广,通过利用改进的初等方法在离散的可测空间中得到了这类新度量,并且证明了得到的新度量成立的充要条件.由此再将这类新度量推广到连续的可测空间中,得到了同样的结果.最后讨论了新度量的最值问题.     

Multivariate Dispersion Models

We introduce a class of multivariate dispersion models suitable as error distributions for generalized linear models with multivariate non-normal responses. The models preserve some of the main properties of the multivariate normal distribution, and include the elliptically contoured distributions and certain other known distributions as special cases. We give explicit methods for constructing multivariate proper dispersion models. This is exemplified by constructing multivariate gamma, Laplace, hyperbola, and von Mises distributions.     

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

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

Distributions of the location of maximuma and minimuma for diffusions with jumps

The paper deals with methods of computation of distributions of location for maxima and minima for diffusions with jumps. As an example, we obtain explicit formulas for distributions of location for the maximum of the process which is equal to the sum of a Brownian motion and the compound Poisson process. Bibliography: 8 titles.     

一类拟平稳序列极值的极限律

Ｋ．Ｆ．Ｔｕｒｋｍａｎ讨论了一类拟平稳序列最大值的渐近分布。本文利用点过程收全党一理得到水平超出点过程的收敛定理和第ｒ个最大值的渐近分布及前ｒ个最大值的联合渐近分布。     

STRUCTURES OF SYSTEMS WITH EXPONENTIAL LIFE DISTRIBUTIONS

Abstract. Structures of monotone systems and cold standby systems with     

Extreme Values in FGM Random Sequences

We consider the multivariate Farlie–Gumbel–Morgenstern class of distributions and discuss their properties with respect to the extreme values. This class was used to consider dependence in multivariate distributions and their ordering. We show that the extreme values of these distributions behave as if no dependence would exist between its components.     

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.     

On Moments of Folded and Truncated Multivariate Normal Distributions

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.     

More accurate breakdown voltage estimation for the new step-up test method in the gumbel distribution model

The estimation problems for the conventional step-up method (the observed breakdown voltages are not given at all) and the new step-up method (some of the observed breakdown voltages are given) are analyzed when the underlying probability distribution (of breakdown voltage level) is assumed to be gumbel distributions for minima and maxima. The new step-up test method has advantages compared to the conventional method: (1) the confidence intervals of the estimates become smaller and (2) the estimates can be obtained with higher probability. In some case of real step-up breakdown voltage test, a fit of the gumbel distribution to the data case is found to be superior to that of the normal distribution, which suggests the usefulness of the gumbel distribution for the underlying distribution in the step-up breakdown voltage test.