The Asymptotic Distribution Theory of Bivariate Order Statistics

In this paper the limit distribution function (d.f.) of general bivariate order statistics (o.s.) (extreme, intermediate and central) is studied by the notion of the exceedances of levels and characteristic function (c.f.) technique. The advantage of this approach is to give a simple and unified method to derive the limit d.f. of any bivariate o.s. The conditions under which the limit d.f. splits into the product of the limit marginals are obtained. Some illustrative examples are given.     

Regression models for multivariate ordered responses via the Plackett distribution

We investigate the properties of a class of discrete multivariate distributions whose univariate marginals have ordered categories, all the bivariate marginals, like in the Plackett distribution, have log-odds ratios which do not depend on cut points and all higher-order interactions are constrained to 0. We show that this class of distributions may be interpreted as a discretized version of a multivariate continuous distribution having univariate logistic marginals. Convenient features of this class relative to the class of ordered probit models (the discretized version of the multivariate normal) are highlighted. Relevant properties of this distribution like quadratic log-linear expansion, invariance to collapsing of adjacent categories, properties related to positive dependence, marginalization and conditioning are discussed briefly. When continuous explanatory variables are available, regression models may be fitted to relate the univariate logits (as in a proportional odds model) and the log-odds ratios to covariates.     

On multivariate infinitely divisible distributions

Simple conditions are given which characterize the generating function of a nonnegative multivariate infinitely divisible random vector. Necessary conditions on marginals, linear combinations, tail behavior, and zeroes are discussed, and a sufficient condition is given. The latter condition, which is a multivariate generalization of ordinary log-convexity, is shown to characterize only certain products of univariate infinitely divisible distributions.     

Dependence between two multivariate extremes

We extend the characterizations given by Takahashi (1988) for the independence and the total dependence of the univariate marginals of a multivariate extreme value distribution to its multivariate marginals. We also deal with the problem of how to measure the strength of the dependence among multivariate extremes. By presenting new definitions for the extremal coefficient, we propose measures that summarize the dependence between two multivariate extreme value distributions and preserve the main properties of the known bivariate coefficient for two univariate extreme value distributions. Finally, we illustrate these contributions to model the dependence among multivariate marginals with examples.     

Sklar’s theorem obtained via regularization techniques

Sklar’s theorem establishes the connection between a joint d-dimensional distribution function and its univariate marginals. Its proof is straightforward when all the marginals are continuous. The hard part is the extension to the case where at least one of the marginals has a discrete component. We present a new proof of this extension based on some analytical regularization techniques (i.e., mollifiers) and on the compactness (with respect to the L^∞ norm) of the class of copulas.     

Three general multivariate semi-Pareto distributions and their characterizations

Three general multivariate semi-Pareto distributions are developed in this paper. First one—GMP^(k)(III) has univariate Pareto (III) marginals, it is characterized by the minimum of two independent and identically distributed random vectors. Second one—GMSP has univariate semi-Pareto marginals and it is characterized by finite sample minima. Third one—MSP is characterized through a geometric minimization procedure. All these three characterizations are based on the general and the particular solutions of the Euler's functional equations of k-variates.     

Operator-stable distributions and stable marginals

Sharpe has shown that full operator-stable distributions μ on Rⁿ are infinitely divisible and for a suitable automorphism B depending on μ satisfy the relation $\text{μ}{}^{\mathrm{t}}\text{= μt}{}^{\mathrm{?B}}\text{1 δ(b(t))}$ for all t > 0. B is called an exponent for μ. It is proved here that if an operator-stable distribution on Rⁿ has n linearly independent univariate stable marginals, then its exponents are semi-simple operators. In addition necessary and sufficient conditions are given for such a distribution on R² to have univariate stable marginals. The proofs use a hitherto unpublished result of Sharpe's that all full operator-stable distributions are absolutely continuous. His proof is provided here.     

关于任意相依随机序列的完全收敛性(英文)

引入极限对数似然比的概念作为任意随机序列的联合分布与其边缘分布的差异的随机性度量,用概率密度比构造几乎处处收敛的上鞅,在适当的条件下,给出任意随机序列完全收敛的若干定理.     

离散随机序列随机和的一类强偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic se-quences,as a measure of “dissimilarity“ between their joint distributions and the product of theirmarginals,is introduced. Construct a. s. convergence supermartingale by means of truncation methodand under suitable restrict Chung-Teicher type conditions,some strong deviation theorems for arbi-trary discrete stochastic sequence are obtained.     

关于任意非负随机序列随机和的一类随机偏差定理

本文引入任意随机变量序列随机极限对数似然比概念,作为任意相依随机序列联合分布与其边缘乘积分布“不相似”性的一种度量,利用构造新的密度函数方法来建立几乎处处收敛的上鞅,在适当的条件下,给出了任意受控随机序列的一类随机偏差定理.     

A class of bivariate distributions including the bivariate logistic

A univariate logistic distribution can be specified by considering a suitable form for the odds in favor of a failure against survival. This concept is extended to the bivariate case and a class of distributions, indexed by a parameter of association, having given marginals is proposed. Some properties of the proposed class of distributions are studied.     

LIMIT OF ITERATES FOR BERNSTEIN POLYNOMIALS DEFINED ON A TRIANGLE

Let f∈C[0,1],and Bn(f,x) be the a-th Bernstein polynomial associated with function f.ln 1967,the limit of iterates for B.(f,x) was given by Kelisky and Rivlin.After this,Many mathematicians studied and generalized this result.But anyway,all these discussions are only for univariate case ,In this paper,the main contrlbution is that the limit of lterates for Bernstein polynomial defined on a triangle is given completely.     

Limit theorems in the context of multivariate long-range dependence

This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.     

Infinitely divisible measures with independent marginals

Summary Necessary and sufficient conditions are found for a system of marginals of an infinitely divisible measure to be independent. The conditions are stated in terms of the Lévy spectral data of the measure. This result is then applied to the case of independent marginals of operator stable measures.     

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

Summary. V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure and a random (a.s.) linear functional on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of under are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions. Received: 25 July 1995 / In revised form: 20 June 1996     

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.     

Semi-parametric multivariate modelling when the marginals are the same

A model is developed for multivariate distributions which have nearly the same marginals, up to shift and scale. This model, based on “interpolation” of characteristic functions, gives a new notion of “correlation”. It allows straightforward nonparametric estimation of the common marginal distribution, which avoids the “curse of dimensionality” present when nonparametically estimating the full multivariate distribution. The method is illustrated with environmental monitoring network data, where multivariate modelling with common marginals is often appropriate.     

Bayesian Copulae Distributions, with Application to Operational Risk Management

The aim of this paper is to introduce a new methodology for operational risk management, based on Bayesian copulae. One of the main problems related to operational risk management is understanding the complex dependence structure of the associated variables. In order to model this structure in a flexible way, we construct a method based on copulae. This allows us to split the joint multivariate probability distribution of a random vector of losses into individual components characterized by univariate marginals. Thus, copula functions embody all the information about the correlation between variables and provide a useful technique for modelling the dependency of a high number of marginals. Another important problem in operational risk modelling is the lack of loss data. This suggests the use of Bayesian models, computed via simulation methods and, in particular, Markov chain Monte Carlo. We propose a new methodology for modelling operational risk and for estimating the required capital. This methodology combines the use of copulae and Bayesian models.      

Generalized lipschitz conditions and riesz derivatives on the space of bessel potentials Lp α

In the foregoing Note (this Journal Vol.I.p. 75-99) the space of n-dimensional Bessel potentials L^p _x was deseribed in terms of generalized Lipschitz conditions of f or its Riesz transform for 0<∝≦2 The still open case ∝>1 is treated in the first half of this paper, firstly by introducing appropriate iterates of the cited conditions, secondly by using derivatives of f and its Riesz transform, in particular the Laplacian △ and the gradient of the Riesz transformation(▽,R and by applying the former results In Section 6 a definition of a Riesz derivative of order ∝ is given and based upon the concept: Integrate f(m-α)-times in the sense of Riesz and then differentiate [d]m-times (by considering the limit of suitable difference quotients of f). Necessary and sufficient conditions for the existence of these Riesz derivatives are obtained All results also hold in the non-reflexive spaces[d]     

二元广义Weibull模型(英文)

本文研究了一个二元广义Weibull分布模型,其边缘分布分别是一元广义Weibull分布.利用EM算法,得到了未知参数的极大似然估计和观测Fisher信息矩阵.