Positive dependence properties of elliptically symmetric distributions

Let X₁, …, X_p have p.d.f. g(x₁² + … + x_p²). It is shown that (a) X₁, …, X_p are positively lower orthant dependent or positively upper orthant dependent if, and only if, X₁,…, X_p are i.i.d. N(0, σ²); and (b) the p.d.f. of |X₁|,…, |X_p| is TP₂ in pairs if, and only if, In g(u) is convex. Let X₁, X₂ have p.d.f. $\text{f(x}{}_1\text{, x}{}_2\text{) =}\text{|Σ|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{g((x}{}_1\text{, x}{}_2\text{)}\text{Σ}{}^{\mathrm{?1}}\text{(x}{}_1\text{, x}{}_2\text{)′)}$. Necessary and sufficient conditions are given for f(x₁, x₂) to be TP₂ for fixed correlation ?. It is shown that if f is TP₂ for all ? >0. then (X₁, X₂)′ ～ N(0, Σ). Related positive dependence results and applications are also considered.     

Some partial orderings of exchangeable random variables by positive dependence

Some partial orderings of positively dependent exchangeable random variables are introduced. The interrelations among them, the inequalities which follow from them and two models which yield such partial orderings are then discussed. Particular examples include ordering multivariate normal, t, χ², Cauchy, exponential, binomial, Poisson, gamma and Farlie-Gumbel-Morgenstern random vectors. Applications to genetic selection and choice of sampling procedures are given.     

Positive dependence orderings and stopping times

We study the positive dependence of pairs of stochastic processes and examine its relation with the properties of certain stopping times. Some special cases, such as dependent random walks, Gaussian processes and exchangeable sequences of elliptically contoured random variables, are taken into account.     

Bivariate extreme value distributions based on polynomial dependence functions

In this paper, we are concerned with bivariate differentiable models for joint extremes for dependent data sets. This question is often raised in hydrology and economics when the risk driven by two (or more) factors has to be quantified. Here we give a full characterization of polynomial models by means of their dependence function and dependence measure. Copyright © 2006 John Wiley & Sons, Ltd.     

A framework for positive dependence

This paper presents, for bivariate distributions, a unified framework for studying and relating three basic concepts of positive dependence. These three concepts are positive dependence orderings, positive dependence properties and measures of positive dependence. The latter two concepts are formally defined and their properties discussed. Interrelationships among these three concepts are given, and numerous examples are presented.Supported by the National Science Foundation under Grant DMS-8301361.Supported by the Air Force Office of Scientific Research under Contract 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γ_f(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γ_f(x,y) is that the underlying joint density f(x,y) is TP₂ (that is, totally positive of order 2) if and only if . This gives an elegant way to investigate the TP₂ property of any bivariate distribution. For the Saramanov family, the Ali-Mikhail-Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y) to be TP₂. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.     

Further developments on some dependence orderings for continuous bivariate distributions

The dependence orderings, more associated and more regression dependent, due to Schriever (1986, Order Dependence, Centre for Mathematics and Computer Sciences, Amsterdam; 1987, Ann. Statist., 15, 1208–1214) and Yanagimoto and Okamoto (1969, Ann. Inst. Statist. Math., 21, 489–505) respectively, are studied in detail for continuous bivariate distributions. Equivalent forms of the orderings under some conditions are given so that the orderings are more easily checkable for some bivariate distributions. For several parametric bivariate families, the dependence orderings are shown to be equivalent to an ordering of the parameter. A study of functionals that are increasing with respect to the more associated ordering leads to inequalities, measures of dependence as well as a way of checking that this ordering does not hold for two distributions.This research has been supported by NSERC Canada grants and a Scientific Grant of the University of Science and Technology of China.     

A concept of negative dependence using stochastic ordering

A concept of negative dependence called negative dependence by stochastic ordering is introduced. This concept satisfies various closure properties. It is shown that three models for negetive dependence satisfy it and that it implies the basic negative orthant inequalities. This concept is also satisfied by the multinomial, multivariate hypergeometric. Dirichlet and Dirichlet compound multinomial distributions. Furthermore, the joint distribution of ranks of a sample and the multivariate normal with nonpositive pairwise correlations also satisfy this condition. The positive dependence analog of this condition is also studied.     

Estimation of a Bivariate Extreme Value Distribution

Several threshold methods have been proposed for the purpose of estimating a bivariate extreme value distribution from a sample of data whose distribution is only in its domain of attraction. An integrated view of these methods is presented which leads to the introduction of a new asymptotically consistent estimator of the dependence function characterizing the extreme dependence structure. Through Monte Carlo simulations, the new estimator is also shown to do as well as its competitors and to outperform them in cases of weak dependence. To the authors' knowledge, this is the first time that the small-sample behavior of nonparametric bivariate threshold methods has ever been investigated.     

A Directory of Coefficients of Tail Dependence

Models characterizing the asymptotic dependence structures of bivariate distributions have been introduced by Ledford and Tawn (1996), among others, and diagnostics for such dependence behavior are presented in Coles et al. (1999). The following pages are intended as a supplement to the papers of Ledford and Tawn and Coles et al. In particular we focus on the coefficient of tail dependence, which we evaluate for a wide range of bivariate distributions. We find that for many commonly employed bivariate distributions there is little flexibility in the range of limiting dependence structure accommodated. Many distributions studied have coefficients of tail dependence corresponding to near independence or a strong form of dependence known as asymptotic dependence.     

Approximations for Bivariate Extreme Values

We study the tail behavior of distributions in the domain of attraction of bivariate extreme value distributions (this includes bivariate extreme value distributions themselves). We provide results on finite approximations of the tail behavior and its analytical shape. The results could form a basis to improve current statistical modeling of bivariate extreme values.     

二元函数极值问题的新评注

对二元函数的极值判定条件进行了新的补充分析,给出了临界情形下的又一充分条件,并做了简明的证明.     

Weak association of random variables

The dependence concept of weak association is introduced and is shown to be equivalent to positive quadrant dependence. Furthermore, a characterization of independence in the class of positive quadrant dependent random variables by means of moment conditions is proved. Both results generalize some theorems proved by Lehmann and Jogdeo for the two- and three-dimensional case.     

The law of the iterated logarithm for positively dependent random variables

The Levy's type maximal inequality is a key to establish the law of the iterated logarithm for associated random variables. Unfortunately, this type inequality cannot be obtained for a generalization of association, i.e., linear positive quadrant dependence, because of their special dependence structure. The purpose of this paper is to provide a different approach to obtain a law of the iterated logarithm for a sequence of linear positive quadrant dependent random variables.     

On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

The paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987,Stochastic Process. Appl.2583–93], the criteria are easily applicable even when the marginal tails are not Pareto-like.     

Positive solutions in the sense of distributions of singular boundary value problems

We obtain positive solutions in the sense of distributions of singular boundary value problems using perturbation and variational methods.

     

ALMOST SURE CONVERGENCE OF THE STABLE TAIL EMPIRICAL DEPENDENCE FUNCTION IN MULTIVARIATE EXTREME STATISTICS

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionandTheoremsSupposethatF(x,y)isabivariatedistributionfunctionwithtwocontinuousmarginaldistributionfunctions,say,FIandF2.DefineFissaidtohaveastabletaildependencefunction(STDF)l(x,y)ifforx20andy20,whereF(x,y)~1--F(QI(x),QZ(y)).TheconceptofSTDFwasintroducedin[6].Supposethat{(Xi,K),i21}isasequenceofi.i.d.randomvectorswithdistributionF(x,y).Ifthereedestsomesequencesofconstantsan>0,on>0,b.ERandd.ER,n>1.suc…     

分数阶微分方程积分边值问题多个正解的存在性

研究了带有积分边值条件的分数阶微分方程的边值问题.利用LeggettWilliams不动点定理,以及一些分析技巧得到了这类分数阶微分方程边值问题多个正解的存在性.     

一类二阶常微分方程组边值问题的三个正解

在边值条件u1(0)=u′1(1)=u2(0)=u′2(1)=0下,研究并得到了二阶常微分方程组