Concepts Of Positive Dependence Research Articles

Some Results on Measures of Interaction among Risks

It has become a common understanding that financial risk can spread rapidly from one institution to another, and the stressful status of one institution may finally result in a systemic crisis. One popular method to assess and quantify the risk of contagion is employing the co-risk measures and risk contribution measures. It is interesting and important to understand how the underlining dependence structure and magnitude of random risks jointly affect systemic risk measures. In this paper, we mainly focus on the conditional value-at-risk, conditional expected shortfall, the delta conditional value-at-risk, and the delta conditional expected shortfall. Existing studies mainly focus on the situation with two random risks, and this paper makes some contributions by considering the scenario with possibly more than two random risks. By employing the tools of stochastic order, positive dependence concepts and arrangement monotonicity, several results concerning the usual stochastic order, increasing convex order, dispersive order and excess wealth order are presented. Concisely speaking, it is found that for a large enough stress level, a larger random risk tends to lead to a more severe systemic risk. We also performed some Monte Carlo experiments as illustrations for the theoretical findings.

SOME PROBABILISTIC INEQUALITIES FOR FUZZY RANDOM VARIABLES

In this paper, the concepts of positive dependence and linearlypositive quadrant dependence are introduced for fuzzy random variables. Also,an inequality is obtained for partial sums of linearly positive quadrant depen-dent fuzzy random variables. Moreover, a weak law of large numbers is estab-lished for linearly positive quadrant dependent fuzzy random variables. Weextend some well known inequalities to independent fuzzy random variables.Furthermore, a weak law of large numbers for independent fuzzy random vari-ables is stated and proved.

Multivariate conditional hazard rate functions – an overview

In this paper, we describe the usefulness and the applications of the multivariate conditional hazard rate functions. First, we define these, as well as the accumulated hazard functions, and then give some properties of them. Using these definitions and properties, we describe the total hazard construction and its main traits. Using the technical tools described previously, we define and discuss various stochastic orders, various positive dependence concepts, and various aging notions that entail nonnegative multivariate random vectors. Copyright © 2014 John Wiley & Sons, Ltd.

SOME NEW APPLICATIONS OF P–P PLOTS

The P–P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P–P order based on P–P plots. For a pair of random variables (X1, Y1) and (X2, Y2) one can see the relative precedence of Y2 over X2 versus that of Y1 over X1 using P–P order. We show that several seemingly very technical and difficult concepts like convex transform order and super-additive ordering can be easily explained with the help of this new partial order. Several concepts of positive dependence can also be expressed in terms of P–P orders of the conditional distributions.

Multiple Point Hypothesis Test Problems and Effective Numbers of Tests for Control of the Family-Wise Error Rate

Abtsrcat We consider a specific class of multiple test problems, consisting of M simultaneous point hypothesis tests in local statistical experiments. Under certain structural assumptions the global hypothesis contains exactly one element ϑ* (say), and ϑ* is least favourable parameter configuration with respect to the family-wise error rate (FWER) of multiple single-step tests, meaning that the FWER of such tests becomes largest under ϑ*. Further-more, it turns out that concepts of positive dependence are applicable to the involved test statistics in many practically relevant cases, in particular, for multivariate normal and chi-square distributions. Altogether, this allows for a relaxation of the adjustment for multiplicity by making use of the intrinsic correlation structure in the data. We represent product-type bounds for the FWER in terms of a relaxed Šidák-type correction of the overall significance level and compute “effective numbers of tests”. Our methodology can be applied to a variety of simultaneous inference problems for multi-dimensional (location) parameters, as frequently occurring in analysis of variance models or in the context of simultaneous categorical data analysis. For example, simultaneous chi-square tests for association of categorical features are ubiquitous in genetic association studies. In this type of model, Moskvina and Schmidt (2008) gave a formula for an effective number of tests utilizing Pearson's haplotypic correlation coeffcient as a linkage disequilibrium measure. Their result follows as a corollary from our general theory and will be gen- eralized.

A study on LTD and RTI positive dependence orderings

In this paper we discuss the properties of the orderings of positive dependence introduced by Hollander et al. [Hollander, M., Proschan, F., Sconing, J., 1990. Information, censoring, and dependence. In: Topics in Statistical Dependence (Somerset, PA, 1987). In: IMS Lecture Notes Monogr. Ser., vol. 16. pp. 257–268. Inst. Math. Statist., Hayward, CA] as generalizations of the bivariate positive dependence concepts of left-tail decreasing (LTD) and right-tail increasing (RTI) random vectors studied by Esary and Proschan [Esary, J.D., Proschan, F., 1972. Relationships among some concepts of bivariate dependence. Ann. Math. Statist. 43, 651–655]. We show which of the postulates proposed by Kimeldorf and Sampson [Kimeldorf, G., Sampson, A.R., 1987. Positive dependence orderings. Ann. Inst. Statist. Math. 39 (1), 113–128] for a reasonable positive dependence ordering are satisfied and how the orders can be studied by restricting them to copulas. We also investigate the relationships of these orders with some other orderings which have appeared in the literature and generalize the same notions of positive dependence. Finally, some applications to extreme value bivariate distributions are discussed.

Some notions of multivariate positive dependence

The authors study new notions of positive dependence that are associated to multivariate stochastic orders of positive dependence introduced recently by Colangelo et al. [J. Multivar. Anal., to appear]. In particular, they discuss the relationship of these new notions to other existing concepts of positive dependence.

Some positive dependence stochastic orders

In this paper we study some stochastic orders of positive dependence that arise when the underlying random vectors are ordered with respect to some multivariate hazard rate stochastic orders, and have the same univariate marginal distributions. We show how the orders can be studied by restricting them to copulae, we give a number of examples, and we study some positive dependence concepts that arise from the new positive dependence orders. We also discuss the relationship of the new orders to other positive dependence orders that have appeared in the literature.

Some counterexamples in positive dependence

We provide some counterexamples showing that some concepts of positive dependence are strictly stronger than others. In particular we will settle two questions posed by Pemantle (J. Math. Phys. 41 (2000) 1371) and Pellerey (Personal communication, 2002) concerning respectively association versus weak association, weak association versus supermodular dependence, and supermodular dependence versus positive orthant dependence.

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy≥0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.

Characterizing and Generating Bivariate Empirical Rank Distributions Satisfying Certain Positive Dependence Concepts

Abstract This article introduces an approach for characterizing the classes of empirical distributions that satisfy certain positive dependence notions. Mathematically, this can be expressed as studying certain subsets of the class SN of permutations of 1, …, N, where each subset corresponds to some positive dependence notions. Explicit techniques for it-eratively characterizing subsets of SN that satisfy certain positive dependence concepts are obtained and various counting formulas are given. Based on these techniques, graph-theoretic methods are used to introduce new and more efficient algorithms for constructively generating and enumerating the elements of various of these subsets of SN. For example, the class of positively quadrant dependent permutations in SN is characterized in this fashion.

Canonical expansions, correlation structure, and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.

Concepts of Setwise Dependence

This paper motivates and introduces a number of new concepts of positive dependence among sets of random variables. In particular, setwise positive upper (and lower) orthant dependence, setwise association, and other related concepts are studied and their relationships are explored. These new concepts are applied to various multivariate normal distributions, some of which have both positive and negative covariances. A variety of new and interesting inequalities are also obtained.

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.

Dependence by Total Positivity

A multivariate generalization of Shaked's bivariate families which are dependent by total positivity (DTP) is introduced. Some interrelationships and inequalities are generalized. Monotonicity properties of conditional hazard rate and mean residual life functions of some multivariate DTP families are investigated. Relationships with other positive dependence concepts are given.

A general theory of some positive dependence notions

A general theory of concepts of positive dependence, which are weaker than association but stronger than orthant dependence, is developed. A random vector X is associated if and only if P( X ∈ A ⌢ B) ≥ P( X ∈ A) P( X ∈ B) for all open upper sets A and B. By requiring the above inequality to hold only for some open upper sets A and B various notions of positive dependence which are weaker than association are obtained. First a general theory is given and then the results are specialized to some concepts of a particular interest. Various properties and interrelationships are derived and some applications are discussed.

Some concepts of mult1variate dependence

Several types of positive dependence are shown to be equiv-alent to the concept of a distribution with a density which is TP2 in pairs. Among these is the concept of m∗-positive depend-ence of Alam and Wallenius. Using this result, all relationships among many of the most important concepts of positive dependence are determined. Furthermore, an application of the equivalences of these types of positive dependence yields a results of Ahmed, Langberg, Leon and Proschan.

A Stochastic Ordering Induced by a Concept of Positive Dependence and Monotonicity of Asymptotic Test Sizes

An ordering of distributions related to a concept of positive dependence is studied and stochastic monotonicity with respect to this ordering is established for a wide class of two-sample test statistics. Asymptotic conservativeness of test sizes under certain departures from independence between samples is discussed. For example, if the observations are paired and the joint density is positive semidefinite then tests such as Kolmogorov-Smirnov, $\chi^2$ and Cramer-von Mises, as well as a large class of linear rank tests, are shown to be asymptotically conservative.

Some concepts of positive dependence for bivariate interchangeable distributions

In this work we consider some familiar and some new concepts of positive dependence for interchangeable bivariate distributions. By characterizing distributions which are positively dependent according to some of these concepts, we indicate real situations in which these concepts arise naturally. For the various families of positively dependent distributions we prove some closure properties and demonstrate all the possible logical relations. Some inequalities are shown and applied to determine whether under- (or over-) estimates, of various probabilistic quantities, occur when a positively dependent distribution is assumed (falsely) to be the product of its marginals (that is, when two positively dependent random variables are assumed, falsely, to be independent). Specific applications in reliability theory, statistical mechanics and reversible Markov processes are discussed.

A Concept of Positive Dependence for Exchangeable Random Variables

An $n$-variate distribution function is said to be positive dependent by mixture (PDM) if it is a mixture of independent $n$-variate distributions with equal marginals. PDM distributions arise in various contexts of reliability and other areas of statistics. We give a necessary and sufficient condition, by means of independent random variables, for an $n$-variate distribution function to be PDM. The distributions and the expectations of the order statistics of PDM and of independent $n$-variate distributions which have the same marginals, are compared and the results applied to obtain bounds for the reliability of certain $k$ out of $n$ systems. A characterization of vectors of expectations of order statistics of PDM distribution is shown. Surprisingly many exchangeable distributions are found to be PDM. We prove a closure property of the class of PDM distributions and list some examples.