On the relationships between copulas of order statistics and marginal distributions (original) (raw)
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Copulas for order statistics with prescribed margins
Journal of Multivariate Analysis, 2014
The joint distribution of order statistics is characterized without reference to a parent distribution. To this end, the possible univariate margins of such a distribution are first determined. The class of possible copulas K is then characterized under the assumption of continuous margins through a description of their minimal support. A truncation-based construction of copulas is also proposed. In the bivariate case, conditions are given for the existence and uniqueness of copulas in this class having maximal support set. Algorithms and examples also show the effectiveness of this construction.
A new class of copulas based on order statistics was introduced by Baker (2008). Here, further properties of the bivariate and multivariate copulas are described, such as that of likelihood ratio dominance (LRD), and further bivariate copulas are introduced that generalize the earlier work. One of the new copulas is an integral of a product of Bessel functions of imaginary argument, and can attain the Fr\'echet bound. The use of these copulas for fitting data is described, and illustrated with examples. It was found empirically that the multivariate copulas previously proposed are not flexible enough to be generally useful in data fitting, and further development is needed in this area.
Metrika, 2017
This article is devoted to characterize several ordering properties of the maximum order statistic of heterogenous random variables with an Archimedean copula. Some examples are also included to illustrate the obtained results. Keywords Hazard rate order blue • Reversed hazard rate order blue • Likelihood ratio order • Order statistics • Archimedean copula 1 Introduction and preliminaries Order statistics play a vital role in many areas such as statistics, reliability, risk management, auction theory and many other branches of applied probability. Let X k:n be the kth smallest order statistic among X 1 ,. .. , X n. In the reliability context, a k-outof-n system consists of n components which operates if and only if at least k of its n components work. This means that the lifetime of such a system can be represented by B M.
Dependence Modeling, 2021
As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distrib...
Joint weak hazard rate order under non-symmetric copulas
Dependence Modeling, 2016
A weak version of the joint hazard rate order, useful to stochastically compare not independent random variables, has been recently defined and studied in [4]. In the present paper, further results on this order are proved and discussed. In particular, some statements dealing with the relationships between the jointweak hazard rate order and other stochastic orders are generalized to the case of non symmetric copulas, and its relations with some multivariate aging notions (studied in [2]) are presented. For this purpose, the new notions of Generalized Supermigrative and Generalized Submigrative copulas are defined. Other new results, examples and discussions are provided as well.
Further Results for a General Family of Bivariate copulas
Communications in Statistics Theory and Methods, 2013
A bivariate family of copulas has been initiated by Cuadras-Augé (1981) and Marshall (1996). Recently, Durante (2007) considered this family as a general family of symmetric bivariate copulas indexed by a generator function and studied some of its dependence properties. In this article, we obtain and describe further aspects of dependence for this family. For example, we have proved that the family has positive likelihood ratio dependence structure if and only if the family reduces to some well-known copulas. We also derive several proper forms for the generator function of this family. Considering a multivariate extension of the bivariate family of copulas provided by Durante et al. (2007), some dependence properties are studied. Finally, some positive dependence stochastic orderings for two random vectors having a copula from the proposed families, are discussed.
On a new partial order on bivariate distributions and on constrained bounds of their copulas
Fuzzy Sets and Systems, 2020
In this paper we study the maximal possible difference N of values of a quasi-copula at two different points of the unit square. This study enables us to give upper and lower bounds, called constrained bounds, for quasi-copulas with fixed value at a given point in the unit square, thus extending an earlier result from copulas to quasi-copulas. It turns out that the two bounds are actually copulas. Difference N is also the main tool in exhibiting two new characterizations of quasi-copulas, a major result of this paper, which sheds new light on the subject of copulas as well. Significant applications of our results are also given in the imprecise probability theory, one of the more important non-standard approaches to probability. After a full-scale bivariate Sklar's theorem has been proven under this approach, we want to establish the tightness of its background before moving to the more general multivariate scene. We present an extension of the partial order on quasi-distributions used in the said theorem, i.e., pointwise order with fixed margins, using again the difference N as a main tool. A careful study of the interplay between the order on quasi-distributions and the order on corresponding quasi-copulas that represent them is also given. Due to a recent result that the quasi-copulas obtained via Sklar's theorem in the imprecise setting are exactly the same as the ones in the standard setting, it is not surprising that results on quasi-copulas can shed some light both on open questions in the standard probability theory and in the imprecise probability theory at the same time.
Copulas: A Review and Recent Developments
Stochastic Models, 2006
In this review paper we outline some recent contributions to copula theory. Several new author's investigations are presented brie°y, namely: order statistics copula, copulas with given multivariate marginals, copula representation via a local dependence measure and applications of extreme value copulas.
Copula-based orderings of multivariate dependence
2010
Abstract: In this paper I investigate the problem of defining a multivariate dependence ordering. First, I provide a characterization of the concordance dependence ordering between multivariate random vectors with fixed margins. Central to the characterization is a multivariate generalization of a well-known bivariate elementary dependence increasing rearrangement.